DIRAC OPERATORS IN REPRESENTATION THEORY

Jing-Song Huang and Pavle Pandzic

HKUST, Hong Kong and University of Zagreb, Croatia

Lecture 1. (g,K)-modules

1.1. Lie groups and algebras. A Lie groupG is a group which is also a smooth manifold,in such a way that the group operations are smooth. In more words, the multiplicationmap from G£G into G and the inverse map from G into G are required to be smooth.The Lie algebra g of G consists of left invariant vector fields on G. The left invariance

condition means the following: let lg : G ! G be the left translation by g 2 G, i.e.,lg(h) = gh. A vector field X on G is left invariant if (dlg)hXh = Xgh for all g, h 2 G.g is clearly a vector space. It can be identified with the tangent space to G at the

unit element e. Namely, to any left invariant vector field one can attach its value at e.Conversely, a tangent vector at e can be translated to all other points of G to obtain a leftinvariant vector field.Note that we did not require our vector fields to be smooth; it is however a fact that a

left invariant vector field is automatically smooth.The operation making g into a Lie algebra is the bracket of vector fields:

[X,Y ]f = X(Y f)¡ Y (Xf),

for X,Y 2 G and f a smooth function on G. Here we identify vector fields with derivationsof the algebra C∞(G), i.e., think of them as first order differential operators.The operation [,] satisfies the well known properties of a Lie algebra operation: it is

bilinear and anticommutative, and it satisfies the Jacobi identity:

[[X,Y ], Z] + [[Y, Z], X] + [[Z,X], Y ] = 0

for any X, Y,Z 2 g.The main examples are various matrix groups. They fall into several classes. We are

primarily interested in the semisimple connected groups, like the group SL(n,R) of n£ nmatrices with determinant 1. Its Lie algebra is sl(n,R), consisting of the n £ n matriceswith trace 0. There are also familiar series of compact groups: SU(n), the group ofunitary (complex) matrices, with Lie algebra su(n) consisting of skew Hermitian matrices,and SO(n), the group of orthogonal (real) matrices with Lie algebra so(n) consisting ofantisymmetric matrices. Further examples are the groups Sp(n), SU(p, q) and SO(p, q);they are all defined as groups of operators preserving certain forms.

Typeset by AMS-TEX

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Other classes of Lie groups one needs to study are solvable groups, like the groupsof upper triangular matrices; nilpotent groups like the groups of unipotent matrices andabelian groups like Rn or the groups of diagonal matrices. They are not of our primaryinterest, but they show up as subgroups of our semisimple groups and therefore have tobe understood.One also often considers reductive groups, which include semisimple groups but are

allowed to have a larger center, like GL(n,R) or U(n).The definitions are easiest to formulate for Lie algebras g: define C0g = D0g = g, and

inductively Ci+1g = [g, Cig], Di+1g = [Dig, Dig]. Then g is nilpotent if Cig = 0 for largei, g is solvable if Dig = 0 for large i, g is semisimple if it contains no nonzero solvableideals, and g is reductive if it contains no nonabelian solvable ideals. Abelian means that[, ] = 0.Instead of checking that each of the above mentioned groups is a Lie group, on can

refer to a theorem of Cartan, which asserts that every closed subgroup of a Lie group isautomatically a Lie subgroup in a unique way. Since each of the groups we mentioned iscontained in GL(n,C) we only need to see that GL(n,C) is a Lie group. But GL(n,C) isan open subset of Cn2 , hence is a manifold, and the matrix multiplication and invertingare clearly smooth. Without using Cartan’s theorem, one can apply implicit/inverse func-tion theorems, as each of the above groups is given by certain equations that the matrixcoefficients must satisfy.Let us note some common features of all the mentioned examples:² g is contained in the matrix algebraMn(C), and [X,Y ] = XY ¡Y X, the commutator

of matrices;² G acts on g by conjugation: Ad(g)X = gXg−1. This is called the adjoint action. The

differential of this action with respect to g gives an action of g on itself, ad(X)Y = [X, Y ],which is also called the adjoint action.² There is an exponential map exp : g ¡! G mapping X to eX . This is a local diffeomor-

phism around 0, i.e., sends a neighborhood of 0 in g diffeomorphically onto a neighborhoodof e in G.

1.2. Finite dimensional representations. Let V be a complex n-dimensional vectorspace. A representation of G on V is a continuous homomorphism

π : G ¡! GL(V ).

Any such homomorphism is automatically smooth; this is a version of the already men-tioned Cartan’s theorem.Given a representation of G as above, we can differentiate it at e and obtain a homo-

morphismdπ = π : g ¡! gl(V )

of Lie algebras.An important special case is the case of a unitary representation. This means V has an

inner product and that all operators π(g), g 2 G are unitary. Then all π(X), X 2 g areskew-hermitian.The main idea of passing from G to g is turning a harder, analytic problem of studying

representations of G into an easier, purely algebraic (or even combinatorial in some sense)

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problem of studying representations of g. Actually, since we are only considering complexrepresentations, we can as well complexify g and study representations of gC.

Example. The most basic example and the first one to study is the representations ofg = sl(2,C). There is an obvious basis for g: take

h =

µ1 00 ¡1

¶; e =

µ0 10 0

¶; f =

µ0 01 0

¶.

Then the commutator relations are

[h, e] = 2e; [h, f ] = ¡2f ; [e, f ] = h.

Let V be an irreducible representation of g of dimension n+ 1. Then there is a basis

fvk¯k = ¡n,¡n+ 2, . . . , n¡ 2, ng

of V , such that the action of h is diagonal in this basis:

π(h)vk = kvk.

The integers k are called the weights of V . The operator π(e) raises the weight by 2, i.e.,π(e)vk is proportional to vk+2 (which is taken to be 0 if k = n). The operator π(f) lowersthe weight by 2. There is an obvious ordering on the set of weights (usual ordering ofintegers), and n is the highest weight of V ; this corresponds to the fact that π(e)vn = 0.Here is a picture of our representation V :

v−ne¡!áf

v−n+2e¡!áf

. . .e¡!áf

vn−2e¡!áf

vn

Finally, the actions of e and f are also completely determined: let us normalize the choiceof vk’s by taking vn to be an arbitrary π(h)-eigenvector for eigenvalue n, and then takingvn−2j = π(f)jvn. Then

(*) π(e)vn−2j = j(n¡ j + 1)vn−2j+2.

All of this is easy to prove; we give an outline here and encourage the reader to completethe details.First, it is a general fact that the action of h diagonalizes in any finite dimensional

representation, as h is a semisimple element of g. One can however avoid using this, andjust note that the action of h will have an eigenvalue, say λ 2 C, with an eigenvector vλ.By finite dimensionality, we can take λ to be the highest eigenvalue. For any µ 2 C, let usdenote by Vµ the µ-eigenspace for π(h); of course, Vµ will be zero for all but finitely manyµ.From the commutator relations, it is immediate that π(e) sends ¿Vµ to Vµ+2, while

π(f) sends Vµ to Vµ−2 for any µ. In particular, the sum of all Vµ is g-invariant, hence itis all of V by irreducibility. One now takes a highest weight vector vλ, defines (as above)

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vλ−2j = π(f)jvλ, and then proves the analog of the formula (*) with λ replacing n, byinduction from the commutator relations. From (*), finite dimensionality and irreducibilityit now follows that ¿λ must be a non-negative integer, say n, and that V is the ¿span ofthe vectors vn, vn−2, . . . , v−n.Thus we have described all irreducible finite dimensional sl(2,C)-modules. Other finite

dimensional representations are direct sums of irreducible ones; this is a special case ofWeyl’s theorem, which says this is true for any semisimple Lie algebra g. One way toprove this theorem is the so called unitarian trick of Weyl: one shows that there is acompact group G with the same representations as g. E.g., for sl(2,C), G = SU(2).Now using invariant integration one shows that every representation of a compact groupis unitary. On the other hand, unitary (finite dimensional) representations are easily seento be direct sums of irreducibles; namely, for every invariant subspace, its orthogonal is aninvariant complement.

We now pass on to describe finite dimensional representations of a general semisimpleLie algebra g over C. Instead of just one element h to diagonalize, we can now have abunch of them. They comprise a Cartan subalgebra h of g, which is by definition a maximalabelian subalgebra consisting of semisimple elements. Elements of h can be simultaneouslydiagonalized in any representation; for each joint eigenspace, the eigenvalues for variousX 2 h are described by a functional λ 2 h∗, a weight of the representation under consider-ation. All possible weights of finite dimensional representations form a lattice in h∗, calledthe weight lattice of g. The nonzero weights of the adjoint representation of g on itselfhave a prominent role in the theory; they are called the roots of g, and satisfy a numberof nice symmetry properties. For example, the roots of sl(3,C) form a regular hexagon inthe plane. The roots of sl(2,C) are 2 and ¡2 (upon identifying h∗ = (Ch)∗ with C).One can divide up roots into positive and negative roots, which gives an ordering on h∗R,

the real span of roots, and also a notion of positive (“dominant”) weights. The irreduciblefinite dimensional representations of g are classified by their highest weights. The possiblehighest weights are precisely the dominant weights.If we denote by n+ (respectively n−) the subalgebra of g spanned by all positive (re-

spectively negative) root vectors, then we have a triangular decomposition

g = n− © h© n+.

For example, if g = sl(n,C), one can take h to be the diagonal matrices in g, n+ the strictlyupper triangular matrices and n− the strictly lower triangular matrices.Let V be an irreducible finite dimensional representation with highest weight λ. The

highest weight vector is unique up to scalar, and is characterized by being an eigenvectorfor each X 2 h, with eigenvalue λ(X), and by being annihilated by all elements of n+.We now address the question of “going back”, i.e., getting representations of G from

representations of g which we have just described. This is called “integrating” or “expo-nentiating” representations.It turns out there is a topological obstacle to integrating representations; this can already

be seen in the simplest case of 1-dimensional (abelian) Lie groups. There are two connected1-dimensional groups: the real line R and the circle group T1. Both have the same Liealgebra R.

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Consider the one-dimensional representations of the Lie algebra R; each of them is givenby t 7! tλ for some λ 2 C (we identify 1 £ 1 complex matrices with complex numbers).All of these representations exponentiate to the group R, and give all possible charactersof R:

t7¡! etλ, λ 2 C.However, of these characters only the periodic ones will be well defined on T1, and etλ isperiodic if and only if λ 2 2πiZ.In general, when G is connected and simply connected, then all finite dimensional

representations of g integrate to G. Any connected G will be covered by a simply connectedG, and the representations of G are those representations of G which are trivial on thekernel of the covering.If G is semisimple connected with finite center, then there is a decomposition

G = KP

called the Cartan decomposition; here K is the maximal compact subgroup of G, while Pis diffeomorphic to a vector space. For example, if G = SL(n,R), then K = SO(n), andP consists of positive matrices in SL(n,R); so P is diffeomorphic to the vector space ofall symmetric matrices via the exponential map. It follows that the topology of G is thesame as the topology of K, and a representation of g will exponentiate to G if and only ifit exponentiates to K.

1.3. Infinite dimensional representations. In general, a representation of G is acontinuous linear action on a topological vector space H. Some typical choices for H are:a Hilbert space, a Banach space, or a Frechet space.The first question would be: can we differentiate a representation to get a representation

of g? The answer is: not quite. Actually, g acts, but only on the dense subspace H∞ ofsmooth vectors (a vector v 2 H is smooth if the map g7! π(g)v from G into H is smooth).For semisimple G with maximal compact K, there is a better choice than H∞: we can

consider the dense subspace HK of K-finite vectors in H (a vector v 2 H is K-finite if theset π(K)v spans a finite dimensional subspace of H). One shows that K-finite vectors areall smooth, and so HK becomes a Harish-Chandra module, or a (g, K) module. A vectorspace V is a (g,K)-module if it has:

(1) an action of g;(2) a finite action of K;(3) the two k-actions obtained from (1) and (2) agree.

In case we want to allow disconnected groups, we also need(4) the g-action is K-equivariant, i.e., π(k)π(X)v = π(Ad(k)X)π(k)v, for all k 2 K,

X 2 g and v 2 V .One usually also puts some finiteness conditions, like finite generation, or admissibilitydefined below.Now a basic fact about (unitary) representations of compact groups is that they can be

decomposed into Hilbert direct sums of irreducibles, which are all finite dimensional. Thiscan be proved using the basic facts about compact operators. Thus our H decomposesinto a Hilbert direct sum of K-irreducibles, and Harish-Chandra modules, being K-finite,

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decompose into algebraic direct sums of K-irreducibles. For δ 2 bK, the unitary dual of K,we denote by V (δ) the δ-isotypic component of a Harish-Chandra module, i.e., the largestK-submodule of V which is isomorphic to a sum of copies of δ. We say V is admissible

if each V (δ) is finite dimensional. In other words, every δ 2 bK occurs in V with finitemultiplicity. Harish-Chandra proved that for each unitary representation H of G, the(g,K)-module HK is admissible.Going back from (g,K)-modules to representations ofG is hard. We call a representation

(π,H) of G a globalization of a Harish-Chandra module V , if V is isomorphic toHK . Everyirreducible V has globalizations. In fact there are many of them; one can choose e.g. aHilbert space globalization (not necessarily unitary), or a smooth globalization. There arealso notions of minimal and maximal globalizations. A few names to mention here areHarish-Chandra, Lepowsky, Rader, Casselman, Wallach and Schmid. We will not needany globalizations, but will from now on work only with (g, K)-modules.

1.4. Infinitesimal characters. Recall that a representation of a Lie algebra g on a vectorspace V is a Lie algebra morphism from g into the Lie algebra End(V ) of endomorphismsof V . Now End(V ) is actually an associative algebra, which is turned into a Lie algebraby defining [a, b] = ab¡ ba; this can be done for any associative algebra. What we want isto construct an associative algebra U(g) containing g, so that representations of g extendto morphisms U(g) ¡! End(V ) of associative algebras. The construction goes like this:consider first the tensor algebra T (g) of the vector space g. Then define

U(g) = T (g)/I,

where I is the two-sided ideal of T (g) generated by elements X − Y ¡ Y − X ¡ [X, Y ],X,Y 2 g. It is easy to see that U(g) satisfies a universal property with respect to mapsof g into associative algebras; in particular, representations of g (i.e., g-modules) are thesame thing as U(g)-modules. Some further properties are² There is a filtration by degree on U(g), coming from T (g);² The graded algebra associated to the above filtration is the symmetric algebra S(g);² One can get a basis for U(g) by taking monomials over an ordered basis of g.The last two properties are closely related and are the content of the Poincare-Birkhoff-

Witt theorem. Loosely speaking, one can think of U(g) as “noncommutative polynomialsover g”, with the commutation laws given by the bracket of g. If we think of elements of gas left invariant vector fields on G, then U(g) consists of left invariant differential operatorson G.Here are some benefits of introducing the algebra U(g).

(1) One can use constructions from the associative algebra setting. For example, thereis a well known notion of “extension of scalars”: let B ½ A be associative algebrasand let V be a B-module. One can consider A as a right B-module for the rightmultiplication and form the vector space A−BV . This vector space is an A-modulefor the left multiplication in the first factor. So we get a functor from B-modulesto A-modules. Another functor like this is obtained by considering HomB(A,V );now the Hom is taken with respect to the left multiplication action of B on A (andthe given action on V ), and the (left!) A-action on the space HomB(A,V ) is givenby right multiplication on A.

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(2) Since g is semisimple, it has no center. U(g) however has a nice center Z(g). Itis a finitely generated polynomial algebra (e.g. for g = sl(n,C) there are n ¡ 1generators and their degrees are 2, 3, . . . , n). The importance of the center followsfrom a simple observation that is often used in linear algebra: if two operatorscommute, then an eigenspace for one of them is invariant for the other. Thismeans that we can reduce representations by taking a joint eigenspace for Z(g).

Let us examine the center Z(g) and its use in representation theory in more detail.First, there is an element that can easily be written down; it is the simplest and mostimportant element of Z(g) called the Casimir element. To define it we need a little morestructure theory.There is an invariant bilinear form on g, the Killing form B. It is defined by

B(X,Y ) = tr(adX adY ), X, Y 2 g.

The invariance condition means that for a Lie group G with Lie algebra g, one has

B(Ad(g)X,Ad(g)Y ) = B(X, Y ), g 2 G, X, Y 2 g.

The Lie algebra version of this identity is

B([Z,X], Y ) +B(X, [Z, Y ]) = 0, X, Y, Z 2 g.

Furthermore, B is nondegenerate on g; this is actually equivalent to g being semisimple.In many cases like for sl(n) over R or C, one can instead of B use a simpler form tr(XY ),which is equal to B up to a scalar.There is a Cartan decomposition of g:

g = k© p.

k and p can be defined as the eigenspaces for the so called Cartan involution θ for theeigenvalues 1 respectively ¡1. There is also a Cartan decomposition G = KP on thegroup level that we already mentioned; here K is the maximal compact subgroup of G (ifG is connected with finite center), and k is the Lie algebra of K.Rather than defining the Cartan involution in general, let us note that for all the matrix

examples in 1.1, θ(X) is minus the (conjugate) transpose of X. So e.g. for g = sl(n,R), kis so(n) and p is the space of symmetric matrices in g.Some further properties of the Cartan decomposition: k is a subalgebra, [k, p] ½ p and

[p, p] ½ k. The Killing form B is negative definite on k and positive definite on p.To define the Casimir element, we choose orthonormal bases Wk for k and Zi for p, so

thatB(Wk,Wl) = ¡δkl; B(Zi, Zj) = δij .

The Casimir element is then

Ω = ¡Xk

W 2k +

Xi

Z2i .

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It is an element of U(g), and one shows by an easy calculation (which we leave as an exer-cise) that it commutes with all elements of g and thus is an element of Z(g). Furthermore,it does not depend on the choice of bases Wk and Zi.Assume now X is an irreducible (g, K)-module. Then every element of Z(g) acts on X

by a scalar. For finite dimensional X it is the well known and obvious Schur’s lemma: letz 2 Z(g) and take an eigenspace for z in X. This eigenspace is a submodule, hence has tobe all of X. For infinite dimensional X the same argument is applied to a fixed K-type inX; see [V1], 0.3.19, or [Kn], Ch. VIII, x3.Now all the scalars coming from the action of Z(g) on X put together give a homomor-

phismχX : Z(g) ¡! C

of algebras, which is called the infinitesimal character of X.By a theorem of Harish-Chandra, Z(g) is isomorphic (as an algebra) to S(h)W , the

Weyl group invariants in the symmetric algebra of a Cartan subalgebra h in g. (The Weylgroup W is a finite reflection group generated by reflections with respect to roots.) Thisis obtained by taking the triangular decomposition g = n− © h © n+ mentioned beforeand building a Poincare-Birkhoff-Witt basis from bases in n−, h and n+. We now get alinear map from U(g) into U(h) = S(h) by projecting along the span of all monomials thatcontain a factor which is not in h. This turns out to be an algebra homomorphism whenrestricted to Z(g), and this gives the required isomorphism.Now we can identify S(h) with the algebra P (h∗) of polynomials on h∗, and recall that

any algebra homomorphism from P (h∗) into C is given by evaluation at some λ 2 h∗.It follows that the homomorphisms from Z(g) into C correspond to W -orbits Wλ in h∗.So, infinitesimal characters are parametrized by the space h∗/W . They are importantparameters for classifying irreducible (g,K)-modules. It turns out that for every fixedinfinitesimal character, there are only finitely many irreducible (g,K)-modules with thisinfinitesimal character. More details about Harish-Chandra isomorphism can be found e.g.in [KV].To finish, let us describe an example where it is easy to explicitly write down all irre-

ducible (g,K)-modules. This is the case G = SL(2,R), whose representations correspondto (sl(2,C), SO(2))-modules.To see the action of K = SO(2) better, we change basis of sl(2,C) and instead of h, e, f

used earlier we now use

W =

µ0 ¡ii 0

¶, X =

1

2

µ1 ii ¡1

¶, Y =

1

2

µ1 ¡i¡i ¡1

¶Note that W 2 kC. The elements W,X and Y satisfy the same relations as before:

[W,X] = 2X; [W,Y ] = ¡2Y ; [X,Y ] =W.

Fix λ 2 C and ² 2 f0, 1g. Define an (sl(2,C), SO(2))-module Vλ,² as follows:² a basis of Vλ,² is given by vn, n 2 Z, n congruent to ² modulo 2;² π

µcos θ sin θ¡ sin θ cos θ

¶vn = e

inθvn;

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² π(W )vn = nvn;² π(X)vn =

12(λ+ (n+ 1))vn+2;

² π(Y )vn =12 (λ¡ (n¡ 1))vn−2.

The picture is similar to the one that we had for finite dimensional representations ofsl(2,C), but now it is infinite:

. . .e¡!áf

vn−2e¡!áf

vne¡!áf

vn+2e¡!áf

. . .

Also, note that we changed normalization for the v0ns; now we do not have a natural placeto start, like a highest weight vector, so it is best to make π(X) and π(Y ) as symmetricas possible.The following facts are not very difficult to check:² Vλ,² is irreducible unless λ+ 1 is an integer congruent to ² modulo 2;² The Casimir element Ω acts by the scalar λ2 ¡ 1 on Vλ,²;² If λ = k ¡ 1 where k ¸ 1 is an integer congruent to ² modulo 2, then Vλ,² contains

two irreducible submodules, one with weights k, k + 2, . . . and the other with weights. . . ,¡k ¡ 2,¡k. If k > 1, these are called discrete series representations, as they occurdiscretely in the decomposition of the representation L2(G). The quotient of Vλ,² by thesum of these two submodules is an irreducible module of dimension k ¡ 1. For k = 1 thetwo submodules are called the limits of discrete series, and their sum is all of V0,1.All this can be found with many more details and proofs in Vogan’s book [V1], Chapter

1. Actually, Chapters 0 and 1 of that book contain a lot of material from this lecture(plus more) and comprise a good introductory reading. Other books where a lot about(g,K)-modules can be found are [Kn] and [W].

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Lecture 2. Clifford algebras, spinors and Dirac operators

2.1. Clifford algebras. From now on we will adopt the convention to denote real Liealgebras ¿with the subscript 0 and to refer to complexified Lie algebras with ¿the sameletter but no subscript. For example, g0 = k0 © p0 ¿will denote the Cartan decompositionof the real Lie algebra g0, ¿while g = k© p will be the complexified Cartan decomposition.We saw in 1.4 that Z(g) has an important role: it reduces representations, and defines

a parameter (infinitesimal character) for the irreducible representations. We also definedthe “smallest” non-constant element of Z(g), the Casimir element Ω.

Let us imagine for a moment that Z(g) contains a smaller (degree one) element T ,such that T 2 = Ω. Then T would (in principle) have more eigenvalues than Ω, as twoopposite eigenvalues for T would square to the same eigenvalue for Ω. As a consequence,we would get a better reduction of representations, and infinitesimal character would bea finer invariant. Of course, such a T does not exist; degree one elements of U(g) are theelements of g and g has no center being semisimple.

The idea is then to twist U(g) by a finite dimensional algebra, the Clifford algebra C(p).The algebra U(g) − C(p) will contain an element D (the Dirac element) whose square isclose to Ω− 1.One can define the Clifford algebra C(p) as an associative algebra with unit, generated

by an orthonormal basis Zi of p (with respect to the Killing form B), subject to therelations

ZiZj = ¡ZjZi (i6= j); Z2i = ¡1.(There are variants obtained by replacing the ¡1 in the second relation by 1 or by 1/2.)This definition involves choosing a basis, so let us give a nicer one:

C(p) = T (p)/I,

where T (p) is the tensor algebra of the vector space p and I is the two-sided ideal of T (p)generated by elements of the form

X − Y + Y −X + 2B(X, Y ).

This definition resembles the definition of U(g); the similarity extends to an analog of thePoincare-Birkhoff-Witt theorem. Namely, C(p) inherits a filtration by degree from T (p).The associated graded algebra is the exterior algebra

V(p). One can obtain a basis for

C(p) by taking an (orthonormal) ordered basis Zi for p and forming monomials over it toobtain

Zi1 . . . Zik , i1 < ¢ ¢ ¢ < ik.Note that no repetitions are allowed here, as Z2i is of lower order. Together with the emptymonomial 1, the above monomials form a basis of C(p) which thus has dimension equal to2dim p.

Finally, let us note that one can analogously construct a Clifford algebra for any vectorspace with a symmetric bilinear form.

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2.2. Dirac operator. Using again our orthonormal basis Zi of p, we define the Diracoperator

D =Xi

Zi − Zi 2 U(g)− C(p).

It is easy to show that D is independent of the choice of basis Zi and K-invariant (for theadjoint action of K on both factors).The adjoint action of k0 on p0 gives a map from k0 into so(p0). On the other hand, there

is a well known embedding of so(p0) into C(p0), given by

Eij ¡Eji 7!¡12ZiZj ,

where Eij denotes the matrix with all matrix entries equal to zero except the ij entrywhich is 1. One can check directly that this map is a Lie algebra morphism (where C(p0)is considered a Lie algebra in the usual way, by [a, b] = ab ¡ ba). Combining these twomaps, we get a map α : k0 ¡! C(p0) ½ C(p), and we use it to produce a diagonal embeddingof k0 into U(g)− C(p), by

X7! X − 1 + 1− α(X), X 2 k0.Complexifying this we get a map of k, hence also of U(k) and Z(k) into U(g)− C(p). Wedenote the images by k∆, U(k∆) and Z(k∆) (∆ for diagonal). In particular, we get Ωk∆ ,the image of the Casimir element Ωk of Z(k). Since Ωk = ¡

PkW

2k , we see

Ωk∆ = ¡Xk

(Wk − 1 + 1− α(Wk))2.

Here is now the announced relationship between D2 and the Casimir element Ωg:

Lemma (Parthasarathy).

D2 = ¡Ωg − 1 +Ωk∆ + C,where C is a constant that can be computed explicitly (C = jjρcjj2 ¡ jjρjj2, where ρ is thehalf sum of positive roots and ρc is the half sum of compact positive roots).

We just start the calculation and invite the reader to continue. Using the relations inC(p), we see the left hand side is

D2 =Xi,j

ZiZj − ZiZj = ¡Xi

Z2i − 1 +Xi<j

[Zi, Zj ]− ZiZj .

On the other hand, the right hand side is

(Xk

W 2k − 1¡

Xi

Z2i − 1)¡Xk

(W 2k − 1 + 2Wk − α(Wk) + 1− α(Wk)

2) +C.

One now shows easily thatXi<j

[Zi, Zj ]− ZiZj = ¡2Xk

Wk − α(Wk),

and somewhat less easily thatPk α(Wk)

2 is a constant (actually, exactly the one mentionedabove).

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2.3. Spinors. Let X be a (g,K)-module, so U(g) acts on X. We want to get the Diracoperator to act, so we have to tensor X with a C(p)-module S; then U(g)− C(p) will acton X − S, in particular D will act. Since we also want to stay as close as possible to X,we want to take minimal, i.e., simple S. It turns out there are only one or two choices forS, depending on whether dim p is even or odd. A simple C(p)-module is called the spinmodule, or a space of spinors.Let us first consider the case when dim p is even, say 2r. The construction of S involves

taking a maximal isotropic subspace u of p with respect to B. We can for example constructone such subspace starting from our orthonormal basis Zi of p0 and dividing it into twogroups, Z1, . . . , Zr and Zr+1, . . . , Z2r. Now we can take for u the span of all Zs + iZr+s,s = 1, . . . , r. Clearly, u will be a complementary isotropic subspace (the conjugation iswith respect to p0), and we have

p = u© u.Since the restriction of B to both u and u is zero, it follows that B identifies u with the dualspace u∗ of u. Furthermore, the Clifford algebras over u and u are equal to the exterioralgebras, since B = 0. We take a basis ui of u and the dual basis ui of u. Let u∗ = u1 . . . urbe the corresponding basis element of

Vtopu. We define S to be the left ideal in C(p)

generated by u∗.More explicitly, one can identify

S »=^(u)u∗ »=

^(u),

(since u annihilates u∗). The action of p is now given by

u ¢ (ui1 ^ ¢ ¢ ¢ ^ uis) = u ^ ui1 ^ ¢ ¢ ¢ ^ uisu ¢ (ui1 ^ ¢ ¢ ¢ ^ uis) = ¡2

Xk

B(u, uik)ui1 ^ . . . buik ¢ ¢ ¢ ^ uisfor u 2 u and u 2 u. Namely, u just Clifford multiplies, or equivalently wedge multiplies,from the left, while u has to commute through, and then eventually gets killed uponmeeting u∗.It is quite easy to show that S is a simple C(p)-module, and not too difficult that it is

the only one up to isomorphism. See e.g. [W], 9.2.1.Let us now briefly examine the case when dim p is odd, say 2r+1. We can still consider

maximal isotropic u and u as before, but now there is an additional element Z 2 p,orthogonal to both u and u, and we take it to be of norm 1. We again take S to be

Vu,

and we want to define an action of Z on S. From the relations it follows that Z must bescalar on every ¿homogeneous component of

Vu, and that these scalars must alternate as

the degree changes. To see this, start by observing that all elements of u kill the top wedgeVru, and that there are no other elements of

Vu killed by all elements of u. Since

Vru is

one dimensional, Z must act on it by a scalar. Now use anticommuting of Z and u∗i ’s to seethat Z acts on

Vr−1u by the opposite scalar, etc. From Z2 = ¡1, the scalars by which Z

can act are i and ¡i. We ¿thus have two choices: Z can be i on Veven u and ¡i on Vodd u,or vice versa. This gives us two nonisomorphic simple modules for C(p) in this case. Oneshows by a similar argument as above that these are the only simple C(p)-modules up toisomorphism.

12

2.4. Spin group. Let us again first assume that dim p is even. Let v 2 p0 be of length1, i.e., B(v, v) = 1. Then v is clearly invertible in C(p0), as v

−1 = ¡v. Consider now theaction of v on p0 by conjugation in C(p0):

rv(X) = vXv−1 = ¡vXv, X 2 p0.

If X is orthogonal to v, then v and X anticommute, and hence rv(X) = ¡X. If X isproportional to v, then rv(X) = X. So we see that rv is minus the reflection with respectto the hyperplane orthogonal to v.All v as above generate a subgroup of the invertible elements of C(p0) which we denote

by Pin(p0). The above discussion shows that we have a map

Pin(p0) ¡! O(p0),

which is surjective and has kernel f1,¡1g. The connected component of Pin(p0) is thespin group Spin(p0); it coincides with the products of an even number of vectors v asabove. It is a compact, semisimple group which is a double cover of the group SO(p0).In case dim p is odd, one can do a similar construction. Instead of doing this, let us

give a uniform description of Spin(p0) valid regardless of the parity of dim p (and alsofor forms which are not necessarily positive definite); see [Ko1], p. 282. Let α be theantiautomorphism of C(p) given by the identity on p. Then Spin(p0) is the group ofall even elements g of C(p0) such that gα(g) = 1 and gxα(g) 2 p0 for all x 2 p0. Forg 2 Spin(p0), define T (g) 2 GL(p0) by T (g)x = gxα(g). Then T : Spin(p0) ¡! SO(p0) isa double covering. In particular, Spin(p0) is compact (since B is positive definite on p0).Since Spin(p0) is contained in C(p), it acts on any C(p)-module. For dim p even, there

is only one such simple module, S. Since Spin(p0) consists of even elements of C(p), itpreserves the subspaces

S+ =Veven

u; S− =Vodd

u

of S. These are actually irreducible, and they are called spin representations.For dim p odd, the two spaces of spinors S1 and S2 are irreducible for Spin(p0) and

equivalent to each other. This representation is also called the spin representation.

2.5. Dirac cohomology. We consider the spin double cover K of K, constructed fromthe following pullback diagram:

K ¡¡¡¡! Spin(p0)y yK ¡¡¡¡! SO(p0)

Now if X is a (g,K)-module, then K acts on X − S by acting on both factors: on Xthrough K and on S through Spin(p0). Moreover, it is easy to show that X − S is a(U(g)− C(p), K)-module. Such modules are defined analogously as (g,K)-modules; hereK acts on U(g)−C(p) through the adjoint action of K on both factors, and the Lie algebra

of K embeds into U(g)− C(p) as the diagonal k∆ described earlier.

13

Dirac operator D acts on X −S and we define the Dirac cohomology of X to be the Kmodule

HD(X) = KerD/KerD \ ImD.If we supposeX is a unitary (g,K)-module, then we can define a positive definite hermitianform such that D is symmetric with respect to this form. For this we use the usual form onS for which all elements of p0 are skew-hermitian (see [W], 9.2.3, or [Ch]). It now followsthat if X is unitary, then KerD\ ImD = 0, and the Dirac cohomology of X is just KerD.More or less all of the above can be found in any of the following references: [Ch], [Ko1],[W] or [Kn] (even case).

14

Lecture 3. Dirac operators and group representations

3.1. Paul Dirac and his operator. Paul Dirac (1902-1984) was one of the greatestphysicists of the 20th century. His research interests were mainly quantum mechanics andelementary particles. A free particle T in R3 is described by a state function ψ(t, x) witht 2 R and x 2 R3. To understand this function, one needs to understand the square rootof the wave operator

¤ = ∂2

∂x20¡ ∂2

∂x21¡ ∂2

∂x22¡ ∂2

∂x23.

In 1928, Dirac found the square root D of ¤. It is a matrix valued first order differentialoperator, and it is now called the Dirac operator.

3.2. Square root of Laplace operator. We first describe the square root of the Laplaceoperator in Rn:

∆ = ¡ ∂2

∂x21¡ ∂2

∂x22¡ ¢ ¢ ¢ ¡ ∂2

∂x2n.

For n = 1, the Laplacian operator has the form ∆ = ¡ ∂2

∂x2 and the Dirac operator

D = i ∂∂x acts on the space of smooth functions f :R! C.For n = 2, the Laplacian operator has the form ∆ = ¡ ∂2

∂x2 ¡ ∂2

∂y2 . We define

D =

µ0 ii 0

¶∂

∂x+

µ0 1¡1 0

¶∂

∂ + y= γx

∂

∂x+ γy

∂

∂y,

which acts on the space of smooth functions f :R2 ! C2. It follows from

γ2x = γ2y = ¡I, γxγy + γyγx = 0

that

D2 = ∆

µ1 00 1

¶= ∆I.

For n = 3, the Laplacian operator has the form ∆ = ¡ ∂2

∂x21¡ ∂2

∂x22¡ ∂2

∂x23. We define

γ1 =

µi 00 ¡i

¶, γ2 =

µ0 ¡11 0

¶, γ3 =

µ0 ii 0

¶.

Then one hasγ2i = ¡I, γiγj + γjγi = 0(i6= j), i, j 2 f1, 2, 3g.

Set

D = γ1∂

∂x1+ γ2

∂

∂x2γ3

∂

∂ + x3,

which acts on the space of smooth functions f :R3 ! C2. It follows that

D2 = ∆

µ1 00 1

¶= ∆I.

15

3.3. The original Dirac operator and its generalizations. Dirac defined the squareroot of the wave operator ¤ in terms of Pauli matrices:

σ0 =

µ1 00 1

¶,σ1 =

µ1 00 ¡1

¶,σ2 =

µ0 ¡ii 0

¶, σ3 =

µ0 11 0

¶.

Then one has

σ2i = I, σjσk = ¡σkσj = ¡iσl, if j < k and fj, k, lg = f1, 2, 3g.

Define 4£ 4 matrices

γ0 =

µ0 σ0σ0 0

¶, γj =

µ0 σj¡σj 0

¶, j = 1, 2, 3

and

D = γ0∂

∂x0+ γ1

∂

∂x1γ2

∂

∂x2+ γ3

∂

∂x3.

It follows from

γ20 = I, γ21 = γ22 = γ23 = ¡I, γjγk = ¡γkγj (j6= k)

thatD2 = ¤I.

Brauer and Weyl afterwards generalized the definition of Dirac operator to arbitraryfinite-dimensional (quadratic) space of arbitrary signature. The Dirac operator definedin 2.2 is a much more recent analogue for semisimple Lie algebras; it was introduced byParthasarathy in 1972.

3.4. Group representations. Representations of finite groups were studied by Dedekind,Frobenius, Hurwitz and Schur at the beginning of the 20th century. In 1920’s, the focus ofinvestigations was representation theory of compact Lie groups and its relations to invari-ant theory. Cartan and Weyl obtained the well-known classification of equivalence classesof irreducible unitary representations of compact Lie groups in terms of highest weights. In1930’s, Dirac and Wigner started the investigation of infinite-dimensional representationsof noncompact Lie groups.Harish-Chandra (1923-1983) was a Ph.D. student at the University of Cambridge under

supervision of Dirac during the years 1945-1947. It was Harish-Chandra who began asystematical investigation of infinite-dimensional representations of semisimple Lie groupsafter his Ph.D. thesis. He lay down the foundation for further development of the theoryfor the last half century. In 1964 and 1965, Harish-Chandra published two papers whichgave a complete parametrization of discrete series representations. Later he used thisclassification to prove the Plancherel formula. This classification of discrete series is alsocrucial to Langlands classification of admissible representations. However, Harish-Chandradid not give explicit construction of discrete series. His work was parallel to that of Cartan-Weyl for irreducible unitary representations of compact Lie groups.

16

In 1955, Borel and Weil gave explicit realization of irreducible unitary representationsof compact Lie groups. In 1957, Bott generalized Borel-Weil theorem by considering Dol-beault cohomology of line bundles over G/T , where T is a maximal torus. In 1967, Schmidproved in his thesis the conjecture of Kostant and Langlands that discrete series repre-sentations are equivalent to certain Dolbeault cohomology of line bundles over G/T fornoncompact G.After Hotta’s work on construction of holomorphic discrete series and Parthasarathy’s

construction of most discrete series representations by Dirac operators, Atiyah and Schmidin 1977 proved that all discrete series can be constructed as kernels of the Dirac operatorover twisted spinor bundles.The definition of the Dirac operator for a spin bundle was a major accomplishment of

Atiyah-Singer, who obtained the celebrated index theorem. The Dirac cohomology is afar reaching generalization of the idea of index theory to representation theory. In therest of the lectures we see how this simplifies proofs of many classical results such as Bott-Borel-Weil theorem and Atiyah-Schmid theorem and how it sharpens a result of Langlandsand Hotta-Parthasarthy on multiplicities of automorphic forms. An important tool in ourapproach is the theory of Aq(λ)-modules which we review in the next lecture. One shouldbear in mind that this theory was not available at the time when the above mentionedclassical results were proved.

17

Lecture 4. Introduction to Aq(λ)-modules

Mackey’s construction of induction is based on real analysis, and Zuckerman’s con-struction of induction which is needed for Aq(λ)-modules is based on complex analysis. Itamounts to the geometric construction of representations by using Dolbeault cohomologysections of vector bundles over a noncompact complex homogeneous spaces then passingto the Taylor coefficients.

4.1. θ-stable parabolic subalgerbas. Let H = TA be a fundamental Cartan in G. Leth0 = t0+a0 be the corresponding θ-stable Cartan subalgebra. Then t0 = h0\k0 is a Cartansubalgebra of k0. As usual we drop the subscript 0 for the complexified Lie algebras. LetX 2 it0 be such that ad(X) is semisimple with real eigenvalues. We define(i) l to be the zero eigenspace of ad(X),(ii) u to be the sum of positive eigenspaces of ad(X),(iii) q to be the sum of non-negative eigenspaces of ad(X).

Then q is a parabolic subalgebra of g and q = l+ u is a Levi decomposition. Furthermore,l is the complexification of l0 = q \ g0. We write L for the connected subgroup of G withLie algebra l0. Since θ(X) = X, l,u and q are all invariant under θ, so

q = q \ k+ q \ p.In particular, q \ k is a parabolic subalgebra of k with Levi decomposition

q \ k = l \ k+ u \ k.We call such a q a θ-stable parabolic subalgebra.Let f ½ q be any subspace stable under ad(t). Then there is a subset fα1, . . . ,αrg of t∗

and subspaces fαi of f such that if y 2 t and v 2 fαi , thenad(y)v = αi(y)v.

We write∆(f, t) = ∆(f) = fα1, . . . ,αrg,

the weights or roots of t in f. Here ∆(f) is a set with multiplicities, with αi havingmultiplicity dim fαi . Then if

ρ(f) = ρ(∆(f)) =1

2

Xαi∈∆(fαi)

αi 2 t∗,

we have

ρ(f)(y) =1

2tr(ad(y)jt) (y 2 t).

Fix a system ∆+(l\k) of positive roots in the root system ∆(l\ k, t). (Note that we extendthe meaning of root system to include the zero weights.) Then

∆+(k) = ∆+(l \ k) [∆(u \ k)is a positive root system for t in k.If Z is an (l, L \K)-module, we write Z# for Z − ^topu. We set

pro(Z#) = HomU(q)(U(g), Z#)L∩K−finite and ind(Z#) = U(g)−U(q) Z#.

Then both pro(Z#) and ind(Z#) are (g, L \K)-modules.

18

4.2. Zuckerman and Bernstein functors Γ. Let V be a (g, L \ K)-module. TheZuckerman functor Γ = Γg,Kg,L∩K can be defined as follows (if K is connected): Let Γ(V ) bethe sum of all finite-dimensional k0-invariant subspaces of V such that the k0-action can belifted to K. For a morphism ϕ : V ¡!W of (g, L\K)-modules, let Γ(ϕ) be the restrictionof ϕ to Γ(V ).The functor Γ is not exact, but only left exact. Therefore one also needs to study the

right derived functors Γj . For the study of unitarity, it is useful to consider also a “leftanalog” of Γ, the Bernstein functor Π = Πg,Kg,L∩K and its left derived functors Πj . Π isnot so easily defined directly; however, there is a uniform way to describe Zuckerman andBernstein functors in terms of Hecke algebras.Let R(K) be the space of distributions on K. Let R(g, K) be the space of left and right

K-finite distributions on G with support in K. Then

R(g,K) »= R(K)−U(k) U(g) »= U(g)−U(k) R(K).

Then one has that the Zuckerman functor on V is

Γ(V ) = HomR(g,L∩K)(R(g,K), V )K−finite,

and the Bernstein functor is

Π(V ) = R(g,K)−R(g,L∩K) V.

We setRj(Z) = Γj(proZ#) and Lj(Z) = Πj(indZ#).

4.3. Irreducibility and unitarity of cohomologically induced modules. The her-mitian inner product is not obvious for cohomological parabolic induction. This makesstudying unitarity a very difficult problem for cohomologically induced modules. Never-theless, Vogan proved the following powerful theorem.

Theorem ([V2]). Suppose q is a θ-stable parabolic subalgebra of g and Z is an (l, L\K)-module with infinitesimal character λ. If Z is weakly good (i.e. Rehλ + ρ(u),αi ¸ 0 forany α 2 ∆(u)), then(i) Lj(Z) = Rj(Z) = 0 for j6= s (s = dim u \ k).(ii) Ls(Z) »= Rs(Z).(iii) If Z is irreducible, then Ls(Z) is irreducible or zero.(iv) If Z is irreducible and in addition is good (i.e. Rehλ+ρ(u),αi > 0 for any α 2 ∆(u)),

then Ls(Z) is irreducible and nonzero.(v) If Z is unitary, then Ls(Z) is unitary.

Remark.. There are two dualities:

(Πg,Kg,L∩K)j »= (Γg,Kg,L∩K)2s−j and Πj(W )h = Γj(Wh)

where Wh is the hermitian dual of W .

19

4.4. Aq(λ)-modules. Now we consider Z to be a one-dimensional representation. λ: l!C is called admissible if it satisfies the following conditions:(i) λ is the differential of a unitary character of L

(ii) if α 2 ∆(u), then hα,λjti ¸ 0.Given q and an admissible λ, define

µ(q,λ) = representation of K of highest weight λjt + 2ρ(u \ p). (4.1)

The following theorem is due to Vogan and Zuckerman.

Theorem ([VZ], [V2]). Suppose q is a θ-stable parabolic subalgebra of g and λ: l! C isadmissible as defined above. Then there is a unique unitary (g,K)-module Aq(λ) with thefollowing properties:

(i) The restriction of Aq(λ) to k contains µ(q,λ) as defined in (4.1);

(ii) Aq(λ) has infinitesimal character λ+ ρ;

(iii) If the representation of k of the highest weight δ occurs in Aq(λ), then

δ = µ(q,λ) +X

β∈∆(u∩p)nββ

with nβ non-negative integers. In particular, µ(q,λ) is the lowest K-type of Aq(λ).

We note that the unitarity of Aq(λ) in the above theorem was proved in [V2]. In thecontext of definition of θ-stable parabolic subalgebras, if we take X to be a regular element,then we obtain a minimal θ-stable subalgebra b = h+ n. We call such a subalgebra b a θ-stable Borel subalgebra. The corresponding representation Ab(λ) is called a fundamentalseries representation. It is the (g,K)-module of a tempered representation of G. If Ghas a compact Cartan subgroup, then Ab(λ) is the (g, K)-module of a discrete seriesrepresentation of G. Moreover, all (g,K)-modules of discrete series representations of Gare of this form; this will be important in Lecture 6. For the proof of the main resultof [HP] (Lecture 5) it is only needed that Ab(λ) has infinitesimal character λ + ρ andthe lowest K-type µ(b,λ) = λ + 2ρn, where ρn = ρ(n \ p). These facts are contained inTheorem 4.4.

4.5. Salamanca-Riba’s classification of the unitary dual with strongly regularinfinitesimal characters. Let h be a Cartan subalgebra of g. Given any weight Λ 2 h∗,fix a choice of positive roots ∆+(Λ, h) for Λ so that

∆+(Λ, h) ½ fα 2 ∆(g, h) j RehΛ,αi ¸ 0g.

Set

ρ(Λ) =1

2

Xα∈∆+(Λ,h)

α.

20

Definition. A weight Λ 2 h∗ is said to be real if

Λ 2 it∗0 + a∗0,

and to be strongly regular if it is real and

hΛ¡ ρ(Λ),αi ¸ 0, 8α 2 ∆+(Λ, h).

Salamanca-Riba [SR] proved that if an irreducible unitary (g,K)-moduleX has stronglyregular infinitesimal character then X »= Aq(λ) for some θ-stable parabolic subalgebra qand admissible character λ of L. Moreover, she proved the following stronger theoremwhich was conjectured by Vogan.

Theorem(Salamanca-Riba). Suppose that X is an irreducible unitary (g,K)-modulewith infinitesimal character Λ 2 h∗ satisfying

RehΛ¡ ρ(Λ),αi ¸ 0, 8α 2 ∆+(Λ, h).

Then there exist a θ-stable parabolic subalgebra q = l + u and an admissible character λof L such that X is isomorphic to Aq(λ).

21

Lecture 5. Vogan’s conjecture and its proof

In this lecture we explain Vogan’s conjecture on Dirac cohomology and a proof of thisconjecture. The presentation mostly follows [HP].

5.1. Vogan’s conjecture. Let T be a maximal torus in K, with Lie algebra t0. Let h bethe centralizer of t in g; it is a θ-stable Cartan subalgebra of g containing t. Since h = t©pt,we get an embedding of t∗ into h∗. Therefore any element of t∗ determines a character ofthe center Z(g) of U(g). Here we are using the standard identification Z(g) »= S(h)W viathe Harish-Chandra homomorphism (W is the Weyl group), by which the characters ofZ(g) correspond to the W -orbits in h∗.We fix a positive root system∆+(k, t) for t in k; let ρc = ρ(∆+(k, t)) be the corresponding

half sum of the positive roots. For any finite dimensional irreducible representation (γ, Eγ)of k, we denote its highest weight in t∗ by γ again.Vogan [V3] made a conjecture which was proved as the following theorem:

Theorem [HP]. Let X be an irreducible (g, K)-module, such that the Dirac cohomology

of X is non-zero. Let γ be a K-type contained in the Dirac cohomology. Then theinfinitesimal character of X is given by γ + ρc.

In view of the remarks in 2.5, in caseX is unitarizable, we get the following consequence:

Corollary. Let X be an irreducible unitarizable (g,K)-module, such that KerD 6= 0.

Let γ be a K-type contained in KerD. Then the infinitesimal character of X is given byγ + ρc.

In fact, Vogan first conjectured the above corollary and then he saw that the abovetheorem should be the right generalization to non-unitary representations.

5.2. An algebraic reduction of the conjecture. Vogan further reduced the claim ofhis conjecture to an entirely algebraic statement in the algebra U(g)− C(p). Let us firstrecall that in 2.2 we described a diagonal copy k∆ of k inside U(g)−C(p). U(k∆) and Z(k∆)denote the corresponding universal enveloping algebra and its center. It is easy to see thatthey are also embedded into U(g) − C(p); namely, if u 2 U(k) is a PBW monomial, thenits image in U(g)− C(p) is the sum of u− 1 and terms of the form w − a, with w havingsmaller degree than u.We can now state Vogan’s algebraic conjecture that implies the theorem in 5.1.

Theorem. Let z 2 Z(g). Then there is a unique ζ(z) in the center Z(k∆) of U(k∆), andthere are K-invariant elements a, b 2 U(g)−C(p), such that

z − 1 = ζ(z) +Da+ bD.

To see that this theorem implies the theorem in 5.1, let x 2 (X − S)(γ) be non-zero,such that Dx = 0 and x /2 ImD. Note that both z − 1 and ζ(z) act as scalars on x. Thefirst of these scalars is the infinitesimal character Λ of X applied to z, and the second isthe k-infinitesimal character of γ applied to ζ(z), that is, (γ + ρc)(ζ(z)).On the other hand, since (z− 1¡ ζ(z))x = Dax, and x /2 ImD, it follows that (z − 1¡

ζ(z))x = 0. Thus the above two scalars are the same, i.e., Λ(z) = (γ + ρc)(ζ(z)).

22

In 5.6 we will show that under identifications Z(g) »= S(h)W »= P (h∗)W and Z(k∆) »=Z(k) »= S(t)WK »= P (t∗)WK the homomorphism ζ corresponds to the restriction of polyno-mials on h∗ to t∗. Here the already mentioned inclusion of t∗ into h∗ is given by extendingfunctionals from t to h, letting them act by 0 on a = pt. It follows that Λ = γ + ρc, asclaimed.

5.3. A differential complex induced by Dirac operator. Let us first note that theClifford algebra C(p) has a natural Z2-gradation into even and odd parts:

C(p) = C0(p)©C1(p).

This gradation induces a Z2-gradation on U(g)−C(p) in an obvious way.We define a map d from U(g)−C(p) into itself, as d = d0 © d1, where

d0 : U(g)−C0(p) ¡! U(g)−C1(p)

is given byd0(a) = Da¡ aD, (5.3a)

andd1 : U(g)−C1(p) ¡! U(g)−C0(p)

is given byd1(a) = Da+ aD. (5.3b)

In other words, if ²a denotes the sign of a, that is, 1 for even a and ¡1 for odd a, thend(a) = Da¡ ²aaD (for homogeneous a, i.e., those a which have sign).We will use the formula for D2 from Lemma 2.2, namely

D2 = ¡Ωg − 1 +Ωk∆ + C

to prove that our d induces a differential on the K-invariants in U(g)− C(p).Proposition. Let d be the map defined in (5.3a) and (5.3b). Then(i) d is K-equivariant, hence induces a map from (U(g)− C(p))K into itself.(ii) d2 = 0 on (U(g)−C(p))K .

Proof. (i) is trivial, since D is K-invariant.Let a 2 (U(g)− C(p))K be even or odd. Then

d2(a) = d(Da¡ ²aaD) = D2a¡ ²DaDaD ¡ ²a(DaD ¡ ²aDaD2) = D2a¡ aD2,

since obviously ²aD = ²Da = ¡²a. From the formula for D2 (Lemma 2.2), we see that awill commute ¿with D2 if and only if it commutes with Ωk∆ . If a is K-invariant, then thisclearly holds, as a then commutes with all of U(k∆). ¤Thus we see that d is a differential on (U(g)− C(p))K , of degree 1 with respect to the

above defined Z2-gradation. Note that we do not have a Z-gradation on (U(g) − C(p))Kso that d is of degree 1, i.e., this is not a complex in the usual sense.

23

5.4. Determination of cohomology of the complex. We want to calculate the co-homology of d. Before we state the result, let us note the following:

Proposition. Z(k∆) is in the kernel of d.

Proof. Since D is K-invariant, it commutes with k∆, and thus with U(k∆) and in particularwith Z(k∆). Since Z(k∆) ½ (U(g)− C0(p))K , the claim follows. ¤

We now state the following theorem which implies Theorem 5.2.

Theorem. Let d be the differential on (U(g)−C(p))K constructed above. Then Ker d =Z(k∆)© Im d. In particular, the cohomology of d is isomorphic to Z(k∆).The proof uses the standard method of filtering the algebra (the filtration comes from

the usual filtration on U(g)), and then passing to the graded algebra. This graded algebrais of course S(g) − C(p). The analogue of our theorem in the graded setting is easy; thecomplex we get is closely related to the standard Koszul complex associated to the vectorspace p. Namely, the operator d induced by d on S(g)−C(p) is given by supercommutingwith

D =Xi

Zi − Zi,

and one easily calculates that

d(U − Zi1 . . . Zik) = ¡2kXj=1

uZij − Zi1 . . . Zij . . . Zik .

Upon identifying C(p) andV(p) as vector spaces and writing

S(g)− C(p) = S(k)− (S(p)−^(p)),

we see that

d = (¡2)id− dp,where dp is the Koszul differential for the vector space p. In particular, d is a differentialwith cohomolgy S(k)−C, which embeds into S(g)−C(p) by embedding C into S(p)−V(p)as the constants. Passing to K-invariants, we see that

Ker d = S(k)K − 1© Im d

on (S(g)−C(p))K . This is the graded version of our theorem.One can now go back to the original setting by an easy induction on the degree of the

filtration. The main point is that one can reconstruct an element of Z(k∆) from its topterm.We refer the reader to our paper [HP] for the details of the above proof. Let us note a

consequence, which immediately proves Vogan’s conjecture, just put b = a.

24

Corollary. Let z 2 Z(g). Then there is a unique ζ(z) 2 Z(k∆), and there is an a 2(U(g)− C1(p))K , such that

z − 1 = ζ(z) +Da+ aD.

Proof. This follows at once from Theorem 5.4, if we just notice that z − 1 commutes withD (indeed, it is in the center of U(g)− C(p)), and being even, it is thus in Ker d. Hence,it is of the form ζ(z) + d(a) = ζ(z) +Da+ aD.

5.5. Dirac inequality and unitary representations with nonzero Dirac cohomol-ogy. We first indicate how to check if a unitarizable X has non-zero Dirac cohomology.

Proposition. Let X be an irreducible unitarizable (g, K)-module with infinitesimal char-

acter Λ. Assume that X − S contains a K-type γ, i.e., (X − S)(γ)6= 0. Assume furtherthat jjΛjj = jjγ + ρcjj. Then the Dirac cohomology of X, KerD, contains (X − S)(γ).Proof. This again uses the formula for D2 from Lemma 2.2. The formula implies that D2

acts on (X − S)(γ) by the scalar

¡(jjΛjj2 ¡ jjρjj2) + (jjγ + ρcjj2 ¡ jjρcjj2) + (jjρcjj2 ¡ jjρjj2) = 0.

It follows from self-adjointness of D that D = 0 on (X − S)(γ). ¤Note that Corollary 5.1 implies the converse of the above proposition: if X is irreducible

unitarizable, with Dirac cohomology containing (X − S)(γ), then the infinitesimal char-acter of X is Λ = γ + ρc. Hence jjΛjj = jjγ + ρcjj. We note that all irreducible unitaryrepresentations with nonzero Dirac cohomology and strongly regular infinitesimal charac-ters were described in [HP]. They are all Aq(λ)-modules. See Proposition 5.6 where thisis explained in a special case.Finally, combining the above proposition with Corollary 5.1, we sharpen the Parthasarathy’s

Dirac inequality:

Theorem (Extended Dirac Inequality). Let X be an irreducible unitarizable (g,K)-module with infinitesimal character Λ. If (X − S)(γ)6= 0, then

jjΛjj · jjγ + ρcjj.

The equality holds if and only if some W conjugate of Λ is equal to γ + ρc.

5.6. Fundamental series and determination of ζ.

Proposition. Let X be an Ab(λ)-module (as in Theorem 4.4) with b a θ-stable Borelsubalgebra, i.e., X is a fundamental series representation. Assume that λja = 0. Then theDirac cohomology of X contains a K-type Eγ of highest weight γ = λ+ ρn.

Proof. The lowest K-type µ(b,λ) has highest weight λ + 2ρn. Since ¡ρn is a weight ofS, Eγ occurs in µ(b,λ) − S, hence in X − S. Since the infinitesimal character of X isλ+ρ = γ+ρc, it follows that Eγ is in the kernel of the Dirac operator D, i.e., in the Diraccohomology of X. ¤Now we can describe the homomorphism ζ explicitly.

25

Theorem. The homomorphism ζ satisfies the following commutative diagram:

Z(g)ζ¡¡¡¡! Z(k)y y

S(h)WRes¡¡¡¡! S(t)WK

Here the vertical arrows are the Harish-Chandra homomorphisms, and the map Res cor-responds to the restriction of polynomials on h∗ to t∗ under the identifications S(h)W =P (h∗)W and S(t)WK = P (t∗)WK . As before, we can view t∗ as a subspace of h∗ by extend-ing functionals from t to h, letting them act by 0 on a.

Proof. Let ζ : P (h∗)W ¡! P (t∗)WK be the homomorphism induced by ζ under the iden-

tifications via Harish-Chandra homomorphisms. Furthermore, let ζ : t∗/WK ¡! h∗/W bethe morphism of algebraic varieties inducing the homomorphism ζ. We have to show thatζ is the restriction map, or alternatively that ζ is given by the inclusion map.We know from the above proposition that the fundamental series representation Ab(λ)

has the lowest K-typeµ(b,λ) = λ+ 2ρn,

and infinitesimal characterΛ = λ+ ρ.

On the other hand, it follows from Proposition 5.5 that if λja = 0, then the Dirac coho-

mology of Ab(λ) contains the K-type of highest weight γ = λ+ ρn.When proving that Theorem 5.2 implies Theorem 5.1, we saw that Λ(z) = (γ+ρc)(ζ(z)),

for all z 2 Z(g). In our present situation we however have

Λ = λ+ ρ = (λ+ ρn) + ρc = γ + ρc,

so it follows that Λ(ζ(z)) = Λ(z) for all z 2 Z(g). This means that ζ(Λ) = Λ, for allinfinitesimal characters Λ of the above fundamental series representations.It is clear that when λ ranges over all admissible weights in h∗ such that λja = 0, then

Λ = λ+ ρ form an algebraically dense subset of t∗. To see this, it is enough to note thatsuch λ span a lattice in t∗. Hence ζ is indeed the inclusion map. ¤5.7. Remark on finite-dimensional representations. Both Vogan and Kostantpointed out to us that Theorem 5.6 can also be proved by considering finite-dimensionalrepresentations with non-zero Dirac cohomology. These are the representations Vλ of high-est weight λ 2 t∗ ½ h∗. See [HP, Remark 5.6.]

26

Lecture 6. Generalized Bott-Borel-WeilTheorem and construction of discrete series

6.1. Kostant cubic Dirac operator. Let G be a compact semisimple Lie group andR be a closed subgroup. Let g and r be the complexifications of the corresponding Liealgebras. Let g = r© p be the orthogonal decomposition with respect to the Killing form.Choose an orthonormal basis Z1, . . . , Zn of p with respect to the Killing form h , i. Kostant[K2] defines his cubic Dirac operator to be the element

D =nXi=1

Zi − Zi + 1− v 2 U(g)−C(p),

where v 2 C(p) is the image of the fundamental 3-form ω 2 V3(p∗),ω(X, Y,Z) = ¡1

2hX, [Y,Z]i

under the Chevalley identificationV(p∗)! C(p). Kostant’s cubic Dirac operator reduces

to the ordinary Dirac operator when (g, r) is a symmetric pair, since ω = 0 for the sym-metric pair. Kostant ([K2, Theorem 2.16]) shows that

D2 = Ωg − 1¡Ωr∆ +C, (6.1)

where C is the constant jjρjj2 ¡ jjρrjj2. This is the generalization of Lemma 2.2. Thesign change comes from the fact that Kostant uses a slightly different definition of C(p),requiring Z2i to be 1 and not -1. Over C, there is no substantial difference between thetwo conventions.Now we can define the cohomology of the complex (U(g)−C(p))R using Kostant’s cubic

Dirac operator exactly as in Lecture 5, i.e., by d(a) = Da ¡ ²aaD. As before, d2 = 0 on(U(g)−C(p))R. Since the degree of the cubic term is zero in the filtration of U(g)−C(p)used in Lecture 5, the proof Theorem 5.4 goes through without change and we get

Theorem. Let d be the differential on (U(g)− C(p))R defined by Kostant’s cubic Diracoperator as above. Then Ker d = Im d © Z(r∆). In particular, the cohomology of d isisomorphic to Z(r∆).

As a consequence we get an analogous homomorphism ζ : Z(g) ! Z(r) for a reductivesubalgebra r in a semisimple Lie algebra g and a more general version of Vogan’s conjecture.

6.2. Kostant’s theorem on cohomology of homogeneous spaces. If we fix a Cartansubalgebra t of r and extend t to a Cartan subalgebra h of g, then ζ is induced by theHarish-Chandra homomorphism exactly as in Theorem 5.6. This was proved in [K3], byconstructing a sufficiently large family of highest weight modules with known infinitesimalcharacters and nonzero Dirac cohomology.Moreover, the homomorphism ζ induces the structure of a Z(g)-module on Z(r), which

has topological significance. Namely, Kostant has shown that from a well-known theoremof H. Cartan [C], which is by far the most comprehensive result on the real (or complex)cohomology of a homogeneous space, one has

27

Theorem [K3]. There exists an isomorphism

H∗(G/R,C) »= TorZ(g)∗ (C, Z(r)).

6.3. A generalized Weyl character formula. We assume that rankR = rankG. LetW 1 ½ Wg be the subset of Weyl group elements that map the positive Weyl chamber forg into the positive Weyl chamber for r. If λ is a dominant weight for g, then λ+ ρ(g) liesin the interior of the Weyl chamber for g. It follows that w(λ+ ρ(g)) lies in the interior ofthe Weyl chamber for r for any w 2 W 1. Thus, w ¦ λ = w(λ+ ρ(g))¡ ρ(r) is a dominantweight for r.

Theorem [GKRS].

Vλ − S+ ¡ Vλ − S− =Xw∈W 1

(¡1)l(w)Uw¦λ.

It follows that

ch(Vλ) =

Pw∈W 1(¡1)l(w)ch(Uw¦λ)Pw∈W1(¡1)l(w)ch(Uw¦0) .

Note that the above formula reduces to the Weyl character formula when R is a maximaltorus T .

6.4. A generalized Bott-Borel-Weil Theorem. We assume that Uµ is an irreducible

representation of R (or R) so that S−Uµ is a representation of R. The Dirac operator actson the smooth and L2-sections on the twisted spinor bundles over G/R, if we let Zi 2 gact by differentiating from the right. So we have

D:L2(G)−R (S − Uµ)! L2(G)−R (S − Uµ).

We write this action in another form:

D: HomR(U∗µ , L

2(G)− S)! HomR(U∗µ , L

2(G)− S).

Then D is formally self-adjoint. By Peter-Weyl theorem, one has L2(G) »= ©λ∈ bGVλ − V ∗λ .It follows that

KerD =Mλ∈ bG

Vλ −KerfD: HomR(U∗µ , V ∗λ − S) ªg.

The proved Vogan’s conjecture implies KerD 6= 0 if and only if λ + ρ(g) is conjugate toµ+ ρ(r) by the Weyl group. Further consideration of the multiplicity results in

Theorem. One has KerD = Vw(µ+ρ(r))−ρ(g) if there exists a w 2Wg so that w(µ+ρ(r))¡ρ(g) is dominant, and KerD is zero if no such w exists.

In the case R = T a maximal torus, this is a version of Borel-Weil theorem.

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Corollary. Consider

D+:L2(G)−R (S+ − Uµ)! L2(G)−R (S− − Uµ)

and the adjoint

D−:L2(G)−R (S− − Uµ)! L2(G)−R (S+ − Uµ).

One has IndexD = dimkerD+¡dimKerD− = (¡1)l(w) dimVw(µ+ρ(r))−ρ(g) if there existsa w 2Wg so that w(µ+ ρ(r))¡ ρ(g) is dominant and it is zero if no such w exists.

6.5. Geometric construction of discrete series. Let G be a linear semisimple non-compact Lie group. Let K be a maximal compact subgroup of G. Assume that rankG =rankK. Let g0 = k0 + p0 be the Cartan decomposition of the Lie algebra of G. Thenu0 = k0 + ip0 is a compact real form of g = g0 −R C. Let U be the compact analyticsubgroup in the complexification GC of G with Lie algebra u0.Borel showed that there exists a torsion free discrete subgroup Γ of G so that ΓnG and

X = ΓnG/K are compact smooth manifolds. For any µ 2 bK, the Dirac operator D actson the smooth sections of the twisted spin bundle in a similar way described in 6.4.

D:C∞(G/K,S − Eµ)! C∞(G/K, S − Eµ).

Note that the above action of D commutes with the left action of G. So we can considerthe elliptic operator

D+µ (X):C

∞(ΓnG/K,S+ −Eµ)! C∞(ΓnG/K,S− − Eµ).

The index of D+µ (X) can be computed by Atiyah-Singer Index Theorem

IndexD+µ (X) =

ZX

f(Θ,Φ),

where Θ is the curvature of X and Φ is the curvature of the twisted spinor bundle overG/K. By the homogeneity, f(Θ,Φ) is a multiple of the volume form depending only on µ,i.e., f(Θ,Φ) = c(µ)dx. Thus

IndexD+µ (X) = c(µ)vol(ΓnG/K).

Let Y = U/K be the compact homogeneous space. By Hirzebruch proportionality princi-ple, the index of

D+µ (Y ):C

∞(U/K,S+ − Eµ)! C∞(U/K, S− −Eµ)

can be computed in the same way and

IndexD+µ (Y ) = (¡1)qc(µ)vol(U/K),

29

where q = dimG/K = dimU/K. It follows that

IndexD+µ (X) = (¡1)q

vol(ΓnG/K)vol(U/K)

IndexD+µ (Y ).

If we normalize the Haar measure so that vol(U) = 1, then

IndexD+µ (X) = (¡1)qvol(ΓnG) IndexD+

µ (Y ).

Let L2(G) »= R bGHj −H∗j dµ(j) be the abstract Plancherel decomposition. Then the L2sections of the twisted spinor bundles are decomposed as

L2(G/K,S − Eµ) »=ZbGHj −HomK(Hj , S − Eµ)dµ(j).

It follows that the L2-sections of KerD are decomposed as

KerD »=ZbGHj −KerfD: HomK(E

∗µ, ,H

∗j − S) ªgdµ(j).

By the proved conjecture of Vogan, if Hj occurs in the decomposition of KerD then ithas infinitesimal character µ+ ρc. There are at most finitely many representations with afixed infinitesimal character. Thus, if KerD is nonzero, the occurred Hj must be in thediscrete spectrum, i.e., a discrete series representation. It follows from Corollary 6.4 andthe above discussion of the indices that w(µ + ρc) ¡ ρ is dominant. In other words, theinfinitesimal character is strongly regular and therefore Hj is an Aq(λ)-module. Write λfor w(µ + ρc). In case λ is regular, this discrete series is isomorphic to Ab(λ) with thelowest K-type w(µ + ρc) + ρc ¡ ρn. Therefore, we proved the following theorem due toAtiyah and Schmid.

Theorem (Atiyah-Schmid [AS]). Let G be a linear group. Then the kernel of the Diracoperator D acting on the L2-sections of the twisted spinor bundle corresponding to Eµ isa discrete series representation if there exists a w 2Wg so that w(µ+ ρc)¡ ρ is dominant,and KerD is zero otherwise.

We note that Atiyah and Schmid also extended this geometric construction of discreteseries to nonlinear groups.

30

Lecture 7. Lie algebra cohomology

7.1. Definition. Let g be a complex Lie algebra and V a g-module. The space ofinvariants of V with respect to g is the vector space

V g = fv 2 V ¯Xv = 0, 8X 2 gg.Note that this definition is consistent with the definition of invariant (fixed) vectors undera group action. Upon differentiation, the condition of being fixed becomes the conditionof being annihilated.One also considers invariants not with respect to whole g but with respect to some

subalgebra. Let us describe an important example. Let g be semisimple, h a Cartansubalgebra, ∆ the set of roots for g with respect to h, ∆+ a set of positive roots, andn = n+ and n− the corresponding subalgebras like in Section 1.2.Then one is interested in the space V n of n-invariants in V , which is now not only a

vector space but an h-module. We already know why this space is interesting; if V is finitedimensional irreducible, then V n is one dimensional (the highest weight subspace), and theh-action on V n, i.e., the highest weight of V , determines V . If V is still finite dimensionalbut reducible, V = ©iVi, then each Vi contributes a weight vector to V n and V n thusencodes the information about this decomposition.We consider the functor V 7! V n from the categoryM(g) of g-modules into the category

V ectC of complex vector spaces. (Analogously, V 7! V n would be a functor from M(g)intoM(h).) This functor is in general left exact, but not exact. This means that if

0 ¡! U ¡! V ¡!W ¡! 0

is a short exact sequence of g-modules (which is a slightly more precise way to writeW = V/U), then

0 ¡! Ug ¡! V g ¡!W g

is exact, but V g ¡! W g is not surjective in general. Thus we define the g-cohomologyfunctors to be the right derived functors of the functor V 7! V g.To motivate the use of derived functors, let us consider the following simple exam-

ple. Knowing how one can benefit from the usual duality operation for vector spaces,V ∗ = HomC(V,C), one would like to have something similar for modules, say over Z (forsimplicity). If one tries to consider V ∗ = HomZ(V,Z), then one notices that for free mod-ules of finite rank it works fine (including double dual giving back the same module), butfor say Z2, one has

HomZ(Z2,Z) = 0.Namely, Z2 is not free, and its generator 1 can not be mapped anywhere but to 0. We canhowever resolve Z2 by free modules:

0 ¡! Z 2·¡! Z ¡! Z2 ¡! 0.

Thus the complex 0 ¡! Z 2·¡! Z ¡! 0, concentrated in degrees ¡1 and 0, should replace themodule Z2. If we take this resolution and plug it in the functor HomZ(¡,Z) instead of themodule Z2 it resolves, we get the complex

0 ¡! Z 2·¡! Z ¡! 0

31

which is concentrated in degrees 0 and 1 (by contravariance, the arrows changed direction).Its zeroth cohomology is 0, corresponding to the observed fact HomZ(Z2,Z) = 0. However,its first cohomology is Z2 (this is Ext1Z(Z2,Z), the first derived functor of Hom). If we nowapply duality again, we get back to our resolution of Z2, that is Z2 in the zeroth cohomology.To make the above completely precise, one would need to pass to derived category,

which is a way of making modules equal (isomorphic) to their resolutions. However, thepoint we wanted to make is visible: the functor itself maybe sometimes gives nothing, butconsidering also the derived functors may give more.We now get back to our covariant, left exact functor V 7! V g. In principle, the right de-

rived functors are defined using injective resolutions 0 ¡! V ¡! I ·; so the i-th g-cohomologyspace of V would be

Hi(g;V ) = Hi((I ·)g).

Namely, as above, we replace V by its resolution, plug the resolution in the functor, geta complex with induced differential, and then take cohomology. However, this is not veryexplicit, as injective resolutions are rather complicated to write down. Fortunately we havea better possibility: note that

V g = Homg(C, V ),

where C is the trivial g-module; simply identify every map φ : C ¡! V with φ(1) 2 V g. Thismeans that our g-cohomology Hi(g;V ) is equal to Extig(C, V ), and to define it (calculateit) we can resolve C by free modules (or projectives) instead of resolving V by injectives.This is done by the so called standard, or Koszul complex

U(g)−V·g ²¡! C ¡! 0,

where the g-action is given by left multiplication in the U(g)-factor, the differential is thedeRham differential

d(u−X1 ^ ¢ ¢ ¢ ^Xk) =Xi

(¡1)i−1uXi −X1 ^ . . . bXi ¢ ¢ ¢ ^Xk+Xi<j

(¡1)i+ju− [Xi,Xj ] ^X1 ^ . . . bXi . . . bXj ¢ ¢ ¢ ^Xk,and ² is the augmentation map, given by 17! 1 and gU(g)7! 0.To see that this is indeed a resolution, one considers the graded version, S(g)−V g and

is thus lead to the analogous question of resolving the trivial module C over a polynomialalgebra. Now for polynomials in one variable, it is obvious how to do this:

0 ¡! C[X] X·¡! C[X] ¡! C ¡! 0

is clearly the required resolution. To increase the number of variables, this is tensored withitself several times; the introduction of signs which leads to the exterior algebra is forcedby the requirement d2 = 0. The reader is invited to try to construct a resolution for twovariables from scratch and see how the exterior algebra appears quite naturally.

32

Now Hi(g;V ) is the i-th cohomology of the complex

Hom·U(g)(U(g)−V·g, V ) = Hom·C(

V·g, V )

with the induced differential

dα(X1 ^ ¢ ¢ ¢ ^Xk) =Xi

Xiα(X1 ^ . . . bXi ¢ ¢ ¢ ^Xk) +Xi<j

(¡1)i+jα([Xi,Xj ] ^ . . . ),

for α 2 Homk−1C (V·g, V ). Of course, H0(g;V ) = V g.

7.2. Properties and basic applications. Here are some properties of these functors:

(1) Any short exact sequence

0 ¡! U ¡! V ¡!W ¡! 0

of g-modules gives rise to a long exact sequence of cohomology

0 ¡! Ug ¡! V g ¡!W g ¡! H1(g;U) ¡! H1(g;V ) ¡! H1(g;W ) ¡! H2(g;U) ¡! . . .

This is a completely general property of derived functors. It can be used to calculatecohomology in some case, or for various proofs, like the one below.

(2) If g is semisimple and if V has nontrivial infinitesimal character, then Hi(g;V ) = 0for all i. This can be proved by noting that since any Z 2 Z(g) with no constantterm vanishes on the trivial module C, then by a standard homological argumentit follows that the action of Z on the standard complex which is a resolution of Cmust be homotopic to 0.

From this statement one can obtain Weyl’s theorem mentioned in 1.2: any finite dimen-sional module over a semisimple Lie algebra is a direct sum of irreducibles. This statementis equivalent to the statement that any short exact sequence

0 ¡! Ui¡! V

p¡!W ¡! 0

of finite dimensional g modules is split, i.e., there is an s : W ¡! V such that p ± s is theidentity on W . Indeed, such a splitting exhibits V as a direct sum of U and W and wecan then decompose modules completely by induction on the dimension.Now to get a splitting, it suffices to know that the sequence

Homg(W,V )p¡! Homg(W,W ) ¡! 0

is exact; then we get our splitting s as a preimage of idW . But the functor Homg(W,¡)is a composition of two exact functors: HomC(W,¡), which is clearly exact, and (¡)gwhich is also exact on finite dimensional modules by the long exact sequence and the factthat H1(g;¡) vanishes on all finite dimensional modules. The vanishing of H1 has to bechecked separately for the trivial module C; this is an easy calculation.

(3) Similarly, Levi’s theorem that any Lie algebra is a semidirect product of the rad-ical and a semisimple subalgebra can be obtained from the fact that the second

33

cohomology of finite dimensional modules over a semisimple Lie algebra vanishes.Namely, by definition the radical r of g is the biggest solvable ideal, so there is anexact sequence

0 ¡! r ¡! g ¡! l ¡! 0

of Lie algebras, with l semisimple. Now the nontrivial extensions of l by r can beseen to correspond to H2(l; r). As the last space is 0, every extension is trivial andhence the above sequence splits.

7.3. Theorems of Casselman-Osborne and Kostant. We next consider a parabolicsubalgebra q of g, with a Levi decomposition

q = l© u;this is a slightly different Levi decomposition from the one mentioned above, with u thenilradical (the largest nilpotent ideal) and l a reductive subalgebra. A special case arisesin the situation already studied; for a Cartan subalgebra h and n coming from a choice ofpositive roots, define

b = h© n.This is a maximal solvable subalgebra of g and such are called Borel subalgebras. Aparabolic subalgebra of g can be defined as any subalgebra containing a Borel subalgebra.One should think of Borel subalgebras as algebras of upper triangular matrices, and ofparabolic subalgebras as algebras of block upper triangular matrices with some fixed shapeof blocks. Then the Levi factor l is the algebra of corresponding block diagonal matrices,and the nilradical u is the algebra of all upper triangular matrices that are zero on blocks.This is for example exactly true (in some complex basis) if g = sl(n,C).We consider the u-cohomology spaces Hi(u;V ) for a g-module V . These spaces are

actually l-modules in a natural way; namely, l acts on the complex defining Hi(u;V ) byacting both on

Vu and on V . This action commutes with the differential, hence descends

to cohomology.Now Z(g), the center of the enveloping algebra of g, acts on the complex Hom·(

Vu, V )

and its cohomology Hi(u;V ) in two ways. One action is through the Harish-Chandrahomomorphism Z(g) ¡! Z(l) and the l (i.e., U(l)) action described above, and the otheris acting on V only. This second action is well defined only for the center and not for therest of U(g).

Casselman-Osborne theorem. The above two actions of Z(g) agree on cohomologyHi(u;V ).

This is a similar statement to Property 2) of 7.2, about the action of the center onthe standard complex which was used to get vanishing of g-cohomology. A proof can befound in [V], Theorem 3.1.5, or [KV], Theorem 4.149. We will also see how to get this ina different way, using Clifford algebra actions and Dirac type operators.

Kostant’s theorem. Let h be a Cartan subalgebra of g and let b = h © n be a Borelsubalgebra corresponding to a choice of positive roots. Let Vλ be the irreducible finitedimensional g-module with highest weight λ. Then

Hi(n;Vλ) =M

w∈W,l(w)=iCw(λ+ρ)−ρ

34

as h-modules. Here Cµ is the one dimensional h-module with weight µ, W is the Weylgroup, and l(w), the length of w 2 W , is the smallest number of simple root reflectionsneeded to write w as their product.

A proof of this statement and also its generalization to the case of u-cohomology canbe found e.g. in [KV], theorems 4.135 and 4.139. The original proof is greatly simplifiedby using the (more recent) Casselman-Osborne theorem to conclude that only the weightsw(λ+ ρ)¡ ρ can appear.Kostant interpreted this theorem as an algebraic version of the Borel-Weil-Bott theorem,

which realizes finite dimensional g-modules as global sections or cohomology of line bundleson the flag variety B of g. (The flag variety consists of all Borel, i.e., maximal solvablesubalgebras of g). He further used it to obtain an algebraic proof of the Weyl characterformula, which explicitly calculates the (global) character of irreducible representations ofcompact groups.

7.4. n-homology and Casselman subrepresentation theorem. Lie algebra homol-ogy is defined in a similar way as cohomology. One starts by defining coinvariants

Vg = V/gV = V −U(g) C

of a g-module V ; this is the largest quotient of V on which g acts trivially. This is a rightexact functor, and Lie algebra homology functors are the left derived functors

Hi(g;V ) = Torgi (C, V ) = Hi(U(g)−

V·g−U(g) V ) = Hi(

V·g− V ).

The differential is again induced by the deRham differential of the standard complex, whichis used to resolve the variable C and thus define the derived functors.To illustrate the importance of (n-)homology, we mention a very standard construction of

representations, namely the real parabolic induction. Start with the Cartan decomposition

g0 = k0 © p0of a real semisimple Lie algebra g0. Let a0 ½ p0 be a maximal abelian subalgebra (sub-space). Then one can consider the (g0, a0)-roots, by the same principle as for h in g (theadjoint action of a0 on g0 diagonalizes etc.) Let n0 be the sum of positive root spaces (fora choice of positive roots). Then one shows that

g0 = k0 © a0 © n0;

this is called the Iwasawa decomposition of g0. There is also a group version of this fact:multiplication defines a diffeomorphism

K £A£N ¡! KAN = G

where A and N are the connected subgroups of G corresponding to a0 and n0. We denoteby L the centralizer of A in G; then L =MA, where M = L\K is the centralizer of A inK. The subgroup

P =MAN

35

is called a minimal (real) parabolic subgroup of G. Note a slight problem with notation:the Lie algebra of P is not p from Cartan decomposition (that one is not a Lie algebra atall). Nevertheless, this is common notation.Here is what is meant by induction from P to G. Let V be a (finite dimensional)

representation of P . Consider the principal bundle G ¡! G/P and construct the associatedG-equivariant vector bundle

G£P VyG/P

The space G£P V consists of classes of the equivalence relation on G£V defined by setting(gp, v) » (g, p ¢ v), g 2 G, p 2 P, v 2 V.

The continuous sections of this bundle form a representation Indc(V ) of G called thecontinuously induced representation. Here G acts on the sections by left translation. (Onecan also consider smooth sections, or L2 sections of this bundle.) The sections can alsobe interpreted as functions from G into V with the appropriate transformation propertyfor P ; in this way one can avoid the language of bundles. We will denote by Ind(V ) the(g,K)-module of K-finite vectors in Indc(V ).Let us consider the case

V = σ − λ− 1,where σ is an irreducible finite dimensional representation of M , λ is a character of A(which is the same as a character of a), and 1 denotes the trivial representation of N . Thisis basically the only case one needs to study. In this case, Ind(V ) is called the principalseries representation.Casselman proved the following version of Frobenius reciprocity: for any (g,K)-module

W ,

Hom(g,K)(W, Ind(V ))∼=¡! Hom(p,M)(W,V ).

Here p is the Lie algebra of P ; note that M is a maximal compact subgroup of P . Sincethe action of n on V is trivial, the second Hom-space is further equal to

Hom(p,M)(W/nW,V ).

If W/nW is not equal to zero, then one can choose V so that the last space is nonzero. Itfollows that W maps nontrivially into Ind(V ), and if W is irreducible this map has to bean embedding. So we get Casselman’s subrepresentation theorem. Namely, W/nW is notequal to zero; in fact, this space contains leading exponents of asymptotic expansions ofmatrix coefficients of W . See [CM].Let us finish this lecture by mentioning briefly two more facts. First, there is a version

of Poincare duality for Lie algebra cohomology. Second, there is a strong relationshipbetween n-cohomology and Beilinson-Bernstein localization theory. In this theory, irre-ducible (g,K)-modules are realized as global sections (or cohomology) of certain sheaveson the flag variety B of g. The relationship is the following: the geometric fiber of thesheaf corresponding to a (g,K)-module V at a point b 2 X is exactly the n-cohomologyof V , where n = [b, b] is the nilradical of b.

36

Lecture 8. (g,K)-cohomology

8.1. Definition. One can study relative Lie algebra cohomology for the pair (g, k). Ithas however become more usual to study pairs (g,K). The main case we are interestedin is the one suggested by our choice of notation: g is a semisimple complex Lie algebraand K is a maximal compact subgroup of a Lie group G with complexified Lie algebra g.This setting can however be generalized; K could be another compact subgroup of G, ora complex reductive subgroup of a complex group with Lie algebra g. In principle K doesnot even have to be reductive, but then it is not as easy as below to write down resolutions.One can also consider similar pairs (A,K) where A is an associative algebra. A could beU(g), or a quotient Uθ corresponding to an infinitesimal character (θ is a Weyl group orbitof an element of h∗).For the purpose of this lecture, let us concentrate on the usual (and the most interesting)

case of (g, K), the first one mentioned above.Formally, (g,K)-cohomology is analogous to g-cohomology. Namely, one can consider

the functor

V 7! V g,K = fv 2 V ¯Xv = 0, kv = v, for all X 2 g, k 2 Kg

of taking (g,K)-invariants. It is a functor from the category M(g,K) of (g, K)-modulesinto the category of complex vector spaces, which is left exact. The (g,K)-cohomologyfunctors V 7! Hi(g,K;V ) are the right derived functors of V 7! V g,K .As before, one can write

V g,K = Hom(g,K)(C, V ),

and thusHi(g,K;V ) = Exti(g,K)(C, V ).

As before, rather than resolving V by injectives, we use a projective resolution of the trivialmodule C. This is the relative standard complex

U(g)−U(k)V·(g/k) ²¡! C ¡! 0;

Since we are considering compact K, we can replace g/k by the K-invariant direct com-plement p. The differential d of the above complex and the map ² are similar as before:

d(u−X1 ^ ¢ ¢ ¢ ^Xk) =Xi

(¡1)i−1uXi −X1 ^ . . . bXi ¢ ¢ ¢ ^Xk+Xi<j

(¡1)i+ju− [Xi,Xj ]p ^X1 ^ . . . bXi . . . bXj ¢ ¢ ¢ ^Xk,for any compact K; here [Xi, Xj ]p denotes the projection of [Xi,Xj ] to p along k. If K isa symmetric subgroup, like in our main case when K is the maximal compact subgroup,then this projection is always zero, so the second sum actually vanishes. ² is as before theaugmentation map, given by 1− 17! 1 and gU(g)− 17! 0.The relative standard complex is obtained from the standard complex for g, by taking

coinvariants with respect to the k-action given by right multiplication on U(g) and adjoint

37

action onV(g), and also with respect to the action of

V(k) on

V(g) by (exterior) multipli-

cation. Exactness is proved using this and exactness of the standard complex for g. Thefact that this is a projective resolution follows from general homological principles: anyfiniteK-module is projective, and the functorW 7! U(g)−U(k)W from finiteK-modules to(g,K)-modules preserves projectives, as it is left adjoint to the exact functor of forgettingthe g-action.Using the above resolution, we can now identify Hi(g,K;V ) with the ith cohomology

of the complex

Hom·(g,K)(U(g)−U(k)V·(p), V ) = Hom·K(V·(p), V ),

with differential

df(X1 ^ ¢ ¢ ¢ ^Xk) =Xi

(¡1)i−1Xi ¢ f(X1 ^ . . . bXi ¢ ¢ ¢ ^Xk)if K is symmetric, and for other K there is another sum, over i < j.There is also a less often mentioned theory of (g,K)-homology. It is constructed by

deriving the functor of (g,K)-coinvariants; so

Hi(g,K;V ) = Tor(g,K)i (C, V ).

which is calculated using the same resolution of C as above.8.2. Applications. A good reference for learning about various applications of (g,K)-cohomology to the theory of automorphic forms is a recent survey article [LS]. Let usmention just one very classical application, the Matsushima formula. Let Γ ½ G be acocompact lattice and let E be a finite dimensional representation of G. Then the groupcohomology of Γ with coefficients in E, which is also equal to the cohomology of the spaceΓnG/K with coefficients in E, can be expressed as

H∗(Γ, E) »=Mπ∈ bG

m(π,Γ)H∗(g,K;Hπ − E),

where m(π,Γ) is the multiplicity of the unitary representation (π, Hπ) of G in L2(ΓnG).

Another application is a construction of derived Zuckerman functors. Namely, let (g, K)be a pair as above, with K complex algebraic (e.g., the complexification of a maximalcompact subgroup). Let T ½ K be a closed reductive subgroup. Let R(K) be the algebraof regular functions on K. Then one can express the derived Zuckerman modules of a(g, T )-module V as

ΓiK,T (V ) = Hi(k, T ;R(K)− V ).

Here the (k, T ) cohomology is taken with respect to the tensor product of the regularaction with the given action on V . K acts by right translation on R(K), and the g-actionis obtained by twisting the given action πV on V : if we regard an element of R(K)−V asa regular function F : K ¡! V , then for X 2 g the function π(X)F is given by

(π(X)F )(k) = πV (Ad(k)X)(F (k)).

There are several versions of this construction, due to Wallach ([W], Chapter 6), Duflo-Vergne, and Milicic-Pandzic.

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8.3. The Vogan-Zuckerman classification. The central result about (g,K)-cohomology(for K the maximal compact subgroup of G) is the classification of irreducible unitary(g,K)-modules with nonzero (g, K)-cohomology obtained by Vogan and Zuckerman in[VZ]. This result can be stated as follows.Let V be irreducible unitary, of the same infinitesimal character as a finite dimensional

representation F . Note that there is only one possible F for any given V . Then V −F ∗ has nonzero (g, K)-cohomology if and only if V is an Aq(λ) module, as described inprevious lectures. (We also saw that in this case the lowest K-type of V gives rise to Diraccohomology.)Note that if V and F do not have the same infinitesimal character then H∗(g, K;V −

F ∗) = 0. In case there is cohomology, it is equal to

HomL∩K(Vi−dim(u∩p)(l \ p),C),

where L is the Levi subgroup involved in the definition of Aq(λ), l is the (complexified)Lie algebra of L and u is the nilradical of q.For more details, see [LS] or [VZ].

39

Lecture 9. Relationship of Diraccohomology to other kinds of cohomology

In this lecture we briefly sketch some results from [HPR], which grew out of the ideasof [V4].

9.1. Kostant’s cubic Dirac operator. In [K2], Kostant has constructed a cubic Diracoperator D corresponding to any decomposition

g = r© s

of a semisimple Lie algebra g with r a reductive subalgebra such that the Killing form isnon-degenerate on r. The definition is as follows: let Zi be an orthonormal basis for s andlet v 2 V3 s correspond to the alternating trilinear form (X,Y,Z)7! B([X, Y ], Z) under

the isomorphism (V3s)∗ »= V3 s induced by the Killing form. ThenD =

Xi

Zi − Zi + 1− v 2 U(g)− C(s).

It is easy to see that D is independent of the choice of basis Zi and r-invariant for theadjoint action. Following Kostant, we now use defining relations Z2i = 1 instead of Z

2i = ¡1

for C(s).Kostant has shown that the formula we had for the Dirac operator corresponding to the

Cartan decomposition, i.e.,

D2 = Ωg − 1¡ Ωr∆ + (jjρjj2 ¡ jjρrjj2)

still holds in this more general situation. The ingredients of this formula are definedanalogously as before.

Applications to u-cohomology. Let us consider a special case when r = l and s = u© ucorrespond to a θ-stable parabolic subalgebra

q = l© u.

Note that u and u are both isotropic for B, that B identifies u with u∗, and that u iscomplex conjugate to u with respect to the real form g0 implicit in the above definitions.Let S =

V·u be a space of spinors for C(s). Since s is even-dimensional, S is unique up

to isomorphism. There are two l-actions on S; one is given by the adjoint action of l onu, and the other is the spin action, coming from l ¡! so(s) ¡! C(s). These two actions arerelated by a twist by ρu; see [K4].Let X be a (g,K)-module. Then the space

X − S = V·u−Xhas an action of D and of the u-homology operator ∂. Moreover, since we can identifyV·

u−X »= Hom(V·u,X)40

using (V·u)∗ »= V·

u∗ »= V·u, we also get an action of the u-cohomology operator d on

X − S.One checks that

D = d+ 2∂

on X − S. To see this, take a basis ui for u and a dual basis u∗i for u; so B(ui, u∗j ) = δij .Now write

D =Xi

u∗i − ui +Xi

ui − u∗i ¡1

4

Xi,j

1− [u∗i , u∗j ]uiuj ¡1

4

Xi,j

1− [ui, uj ]u∗i u∗j .

Denote by C (respectively C−) the element of U(g)− C(s) composed of the first and thethird term (respectively the second and the fourth term) in the above expression for D.Then show that the action of C on X − S induces the differential d, while the action ofC− induces 2∂.The “half Diracs” C and C− are independent of the choice of basis ui and l-invariant.

They however do depend on the choice of u inside s, while D does not. Furthermore, bothC and C− square to zero, and their supercommutator CC− +C−C is equal to D2.It is convenient to introduce another element,

E = ¡12

Xi

1− u∗iui.

This operator acts as a degree operator on X − S. It satisfies the following commutingrelations:

[E,C] = C; [E,C−] = ¡C−; [E,D2] = 0.

It follows that E, C, C− andD2 span a four dimensional superalgebra inside (U(g)−C(s))l.This algebra is actually Z-graded, if we set C− to be of degree ¡1, D2 and E of degree 0and C of degree 1. This grading is compatible with the obvious grading of X − S comingfrom the standard grading of

V·u.

This superalgebra was used by physicists under the name supersymmetric algebra. Itis denoted by l(1, 1) in Kac’s classification [Kac]. It is completely solvable and its repre-sentation theory is easy but not trivial.As an application of the above facts, let us mention that one can easily prove that

the main result of [HP] holds for C and C− and in this way get the Casselman-Osbornetheorem (see 7.3) as a corollary. Thus the formal analogy between the two results becomesmore concrete.There are two questions that arise from the above considerations. The first one, asked

by Vogan in [V4], is to relate the Dirac cohomology, u-cohomology and u-homology of a(g,K)-module X. This could be useful for example to pass between different choices foru within the same s. The second question was implicitly posed by Kostant in [K4]. Hisremark was that X − S can be formed not just in the “Levi factor case”, i.e., for r = l,but also for a wider class of r described above. On X − S there is always a cubic Diracoperator, and its square is the Laplacian. So it looks natural to try and study this moregeneral setting and see how to make use of that.

41

At this moment, we have some results regarding the first question. Namely, in certainspecial cases all three (co)homology modules are equal up to appropriate twists. The twistis coming from the already mentioned identification of spin and adjoint actions on

V(u),

and it is given by ρu.

9.3. Hodge decomposition. Let us take X to be unitary, so it carries an invariantpositive definite hermitian form. We need to put a similar form on S. There is a natural,l-invariant one, given by the Killing form B on

V·u. This form is however indefinite if

u intersects both k and p as will usually be the case. The operators d and 2∂ can beshown to be adjoint with respect to this form, but one can not in general obtain a “Hodgedecomposition”.A good case where one can get the “Hodge theory approach” to work is when l = k so

that u ½ p and B is positive definite on u. This is possible only in the Hermitian symmetriccase (i.e., when k has a center). In this case the Dirac operator D is “ordinary” (i.e., hasno cubic term) - it is the same D we studied in Lecture 5. We know that in this situation

D2 is a scalar on each K-type, so in particular, D2 acts (locally) finitely on X − S. Notealso that D2 is negative semidefinite in the present situation. Namely, the relations in theClifford algebra are now Z2i = 1, not ¡1, hence D is skew-symmetric and not symmetricas before. So all eigenvalues of D2 are non-positive.Because of these facts, one can use the easy variant of Hodge theory for finite dimensional

spaces (see [W], Scholium 9.4.4), and conclude that

KerD = KerD2 = Ker d \Ker∂;

Ker d = KerD © Im d; Ker ∂ = KerD © Im∂.

In particular, the cohomology of both d and ∂ is equal to the Dirac cohomology of D,KerD, as vector spaces. To compare them as l = k-modules involves the above mentionedtwist.The same argument proves an analogous result for X finite dimensional; here one uses

the “admissible form” on X, i.e., the one invariant for the compact form of g. This casewas known to Vogan and it is also implicit in [K4].

9.4. A counterexample. Here is a simple example which shows that the equality of thethree cohomology modules does not hold for general (sl(2,C), SO(2))-modules. Considerthe module V which is a nontrivial extension of the discrete series representation of highestweight ¡2 by the trivial module C:

0 ¡! C ¡! V ¡!W ¡! 0.

V is a submodule of the module V−1,0 from the end of Lecture 1. The weights of V (for

the basis element

µ0 ¡ii 0

¶of k) are ¢ ¢ ¢ ¡ 4,¡2, 0. We are considering the case l = k, u is

spanned by u = X = 12

µ1 ii ¡1

¶and u is spanned by u∗ = Y = 1

2

µ1 ¡i¡i ¡1

¶. Now

V − S = V − 1© V − u,

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with d : V − 1 ¡! V − u given by d(v − 1) = u∗ ¢ v − u, and ∂ : V − u ¡! V − 1 given by∂(v − u) = u ¢ v − 1.By an easy direct calculation one sees that the u-homology of V is given by

H0(∂) = 0; H1(∂) = Cv0 − u,

the u-cohomology of V is given by

H0(d) = Cv0 − 1; H1(d) = Cv0 − u© Cv−2 − u,

and the Dirac cohomology of V is given by

HD(V ) = KerD = Cv0 − u.

So we seeHD(V ) = H·(∂)6= H ·(d).

9.5. Dirac cohomology and (g,K)-cohomology. In the rest of this lecture we makea few comments on the relationship of Dirac cohomology and (g,K)-cohomology. It wasproved in [HP] that if X is unitary and has (g, K)-cohomology, i.e.,

H∗(g,K;X − F ∗) = H∗(Hom·K(V·p,X − F ∗))6= 0

for a finite dimensional F (which then necessarily has the same infinitesimal character asX), then X also has Dirac cohomology.In the following we assume that dim p is even. Then we can write p as a direct sum

of isotropic vector spaces u and u »= u∗. One considers the spinor spaces S =V·u and

S∗ =V·u; then

S − S∗ »= V·(u© u) = V·p.It follows that we can identify the (g, K)-cohomology of X − F ∗ with

H∗(Hom·K(F − S,X − S)).

There are several possible actions of the Dirac operator D on the above complex; similarlyas before, they can be related to the coboundary operator d and the boundary operator ∂for (g,K)-homology, which also acts on the same complex after appropriate identifications.Now if X is unitary, Wallach has proved that d = 0 (see [W], Proposition 9.4.3, or

[BW]). Using similar arguments one can analize the above mentioned Dirac actions andthe actions of the corresponding “half-Diracs”. In particular, it follows that

H∗(g, K;X − F ∗) = Hom·K(HD(F ),HD(X)).

This can be concluded from the fact that the eigenvalues of D2 are of opposite signs onF − S and X − S; see [W], 9.4.6. One may hope to generalize some of these facts, eitherwith respect to X, or with respect to F .

43

We finish by a remark that for nice and natural proofs it would be useful to formalizesome constructions in the category of modules over the Clifford algebra, although this maynot seem necessary as the category is extremely simple. For example, one can construct acoproduct of C(p) by defining

c(Z) =1p2(Z − 1 + 1− Z)

for Z 2 p. This coproduct is an algebra morphism in the graded sense. It is not coasso-ciative, but it is cocommutative. There is no counit, but there is an antiautomorphism,the already mentioned α (see 2.4) which can be used instead of an antipode. It seemsworthwhile to try to use this structure to clarify notions like tensor products and dualmodules.

44

Lecture 10. Multiplicities of automorphic forms

In this lecture we prove a formula for multiplicities of automorphic forms which sharpensthe result of Langlands and Hotta-Parthasarathy. Let G be a linear semisimple noncompactLie group. Let K be a maximal compact subgroup of G. Assume that rankG = rankK.Let g0 = k0 + p0 be the Cartan decomposition of the Lie algebra of G. Then u0 = k0 + ip0is a compact real form of g = g0 −R C. Let U be the compact analytic subgroup in thecomplexification GC of G with Lie algebra u0.

10.1. Hirzebruch proportionality principle. Let Γ be a torsion free discrete subgroupof G so that ΓnG and X = ΓnG/K are compact smooth manifolds. Borel showed thatsuch a Γ always exists. Then the regular representation on ΓnG is decomposed discretelywith finite multiplicities:

L2(ΓnG) »=Mπ∈ bG

m(Γ,π)Hπ.

Let Xπ be the Harish-Chandra module of Hπ. For any µ 2 bK, we define the Diracoperators D, D+

µ (X) and D+µ (Y ) as in 6.5. As we saw in 6.5, if we normalize the Haar

measure so that vol(U) = 1, then

IndexD+µ (X) = (¡1)qvol(ΓnG) IndexD+

µ (Y ).

On the other hand,

IndexD+µ (X) =

Xπ∈ bG

m(Γ, π) IndexD+µ (Xπ),

where D+µ (Xπ) : HomK(E

∗µ, Xπ − S+) ! HomK(E

∗µ,Xπ − S−) is the linear map defined

by φ7! D ± φ for any φ 2 HomK(E∗µ, Xπ − S+).10.2. Dimension of automorphic forms. If IndexD+

µ (Xπ) 6= 0, then the Dirac co-homology HD(Xπ) contains Eµ∗ . It follows from the proved Vogan’s conjecture that theinfinitesimal character of Xπ is given by µ

∗ + ρc. If we assume that λ = w(µ∗ + ρc) ¡ ρ

is dominant for some w 2 W , then Xπ is isomorphic to Aq(λ) for some θ-stable para-bolic subalgebra q. If in addition we assume that λ is regular with respect to the non-compact roots ∆+(p), then Xπ is uniquely determined as a discrete series Ab(λ). SinceIndexD+

µ (Ab(λ)) = dimD+µ (Ab(λ)) ¡ codimD+

µ (Ab(λ)) = (¡1)q and IndexD+µ (Y ) has

been calculated in Corollary 6.4., we obtain the following theorem.

Theorem. Let π = Ab(λ) be a discrete series representation with λ regular with respectto all noncompact roots. Assume that λ is dominant and can be written as λ = µ¡ ρn forsome highest weight µ 2 bK. Then

m(Γ, π) = vol(ΓnG)dπ,where dπ is the formal degree of π:

dπ =Πα∈∆+(g,t)(λ+ ρ,α)

Πα∈∆+(g,t)(ρ,α).

45

This sharpens the result of Langlands [L] and Hotta-Parthasarathy [HoP], who provedthe above formula for discrete series representations whose matrix coefficients are in L1(G).Trombi-Varadarajan proved that if the matrix coefficients of the discrete series Ab(λ) arein L1(G), then

hλ+ ρ,αi >Max f jhwρ,αij,8w 2Wg and 8α 2 ∆+(p) g.

Hecht-Schmid proved this is also a sufficient condition. Our assumption on the regularityof λ with respect to the noncompact roots amounts to the condition

hλ+ ρ,αi >Max f jhwρ,αij,8α 2 ∆+(p) with w = 1. g.

Therefore, our condition is weaker than that assumed by Langlands and Hotta-Parthasarathy.

10.3. A final remark. Borel and Wallach proved that for any finite-dimensional repre-sentation of G,

H∗(Γ, F ) =Mπ∈ bG

m(Γ, π)H∗(g, K,Xπ − F ).

We still assume that rankG = rankK. If the highest weight of F is regular, then it followsfrom a similar argument as in 10.2. that Xπ is uniquely determined as a discrete seriesAb(λ), and therefore,

dimH∗(Γ, F ) = vol(ΓnG)dπ dimH∗(g,K,Xπ − F ).

46

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[KV] A. W. Knapp and D. A. Vogan, Jr., Cohomological induction and unitary representations, Prince-ton University Press, 1995.

[K1] B. Kostant, Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the ρ-decompositionC(g) = EndVρ ⊗ C(P ), and the g-module structure of ∧g, Adv. in Math. 125 (1997), 275-350.

[K2] B. Kostant, A cubic Dirac operator and the emergence of Euler number multiplets of represen-tations for equal rank subgroups, Duke Math. Jour. 100 (1999), 447-501.

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[V4] D. A. Vogan, Jr., n-cohomology in representation theory, a talk at “Functional Analysis VII”,Dubrovnik, Croatia, September 2001..

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